Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1
Holomorphic vector bundles on ℂ × , a complex manifold, with meromorphic connections with poles of Poincaré rank 1 along {0} × , arise naturally in algebraic geometry. They are called ( )-structures here. This paper takes an abstract point of view. It gives a complete classification of all ( )-s...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України
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| Цитувати: | Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1. Claus Hertling. SIGMA 17 (2021), 082, 73 pages |
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| author | Hertling, Claus |
| author_facet | Hertling, Claus |
| citation_txt | Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1. Claus Hertling. SIGMA 17 (2021), 082, 73 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Holomorphic vector bundles on ℂ × , a complex manifold, with meromorphic connections with poles of Poincaré rank 1 along {0} × , arise naturally in algebraic geometry. They are called ( )-structures here. This paper takes an abstract point of view. It gives a complete classification of all ( )-structures of rank 2 over germs ( , ⁰) of manifolds. In the case of , they separate into four types. Those of the three types have universal unfoldings; those of the fourth type (the logarithmic type) do not. The classification of unfoldings of ( )-structures of the fourth type is rich and interesting. The paper finds and lists all ( )-structures which are basic in the following sense: Together they induce all rank 2 ( )-structures, and each of them is not induced by any other ( )-structure in the list. Their base spaces turn out to be 2-dimensional F-manifolds with Euler fields. The paper also provides a classification of all rank 2 ( )-structures over it. Also, this classification is surprisingly rich. The backbone of the paper is normal forms. Though also the monodromy and the geometry of the induced Higgs fields and of the base spaces are important and are considered.
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| first_indexed | 2026-03-15T12:06:11Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 082, 73 pages
Rank 2 Bundles with Meromorphic Connections
with Poles of Poincaré Rank 1
Claus HERTLING
Lehrstuhl für algebraische Geometrie, Universität Mannheim,
B6, 26, 68159 Mannheim, Germany
E-mail: hertling@math.uni-mannheim.de
URL: https://www.wim.uni-mannheim.de/hertling/team/prof-dr-claus-hertling/
Received September 30, 2020, in final form August 20, 2021; Published online September 07, 2021
https://doi.org/10.3842/SIGMA.2021.082
Abstract. Holomorphic vector bundles on C×M , M a complex manifold, with meromor-
phic connections with poles of Poincaré rank 1 along {0} ×M arise naturally in algebraic
geometry. They are called (TE)-structures here. This paper takes an abstract point of view.
It gives a complete classification of all (TE)-structures of rank 2 over germs
(
M, t0
)
of mani-
folds. In the case of M a point, they separate into four types. Those of three types have
universal unfoldings, those of the fourth type (the logarithmic type) not. The classification
of unfoldings of (TE)-structures of the fourth type is rich and interesting. The paper finds
and lists also all (TE)-structures which are basic in the following sense: Together they induce
all rank 2 (TE)-structures, and each of them is not induced by any other (TE)-structure in
the list. Their base spaces M turn out to be 2-dimensional F -manifolds with Euler fields.
The paper gives also for each such F -manifold a classification of all rank 2 (TE)-structures
over it. Also this classification is surprisingly rich. The backbone of the paper are normal
forms. Though also the monodromy and the geometry of the induced Higgs fields and of
the bases spaces are important and are considered.
Key words: meromorphic connections; isomonodromic deformations; (TE)-structures
2020 Mathematics Subject Classification: 34M56; 34M35; 53C07
Contents
1 Introduction 2
2 The two-dimensional F -manifolds and their Euler fields 5
3 (TE)-structures in general 7
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 (TE)-structures with trace free pole part . . . . . . . . . . . . . . . . . . . . . . 10
3.3 (TE)-structures over F -manifolds with Euler fields . . . . . . . . . . . . . . . . . 12
3.4 Birkhoff normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.5 Regular singular (TE)-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.6 Marked (TE )-structures and moduli spaces for them . . . . . . . . . . . . . . . . 17
4 Rank 2 (TE)-structures over a point 20
4.1 Separation into 4 cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 The case (Sem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3 Joint considerations on the cases (Bra), (Reg) and (Log) . . . . . . . . . . . . . . 23
This paper is a contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji
Saito for his 77th birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Saito.html
mailto:hertling@math.uni-mannheim.de
https://www.wim.uni-mannheim.de/hertling/team/prof-dr-claus-hertling/
https://doi.org/10.3842/SIGMA.2021.082
https://www.emis.de/journals/SIGMA/Saito.html
2 C. Hertling
4.4 The case (Bra) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.5 The case (Reg) with trU = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.6 The case (Log) with trU = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5 Rank 2 (TE)-structures over germs of regular F -manifolds 33
6 1-parameter unfoldings of logarithmic (TE)-structuresover a point 36
6.1 Numerical invariants for such (TE )-structures . . . . . . . . . . . . . . . . . . . . 36
6.2 1-parameter unfoldings with trace free pole part of logarithmic pure (TLE)-
structures over a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.3 Generically regular singular (TE)-structures over (C, 0) with logarithmic restric-
tion over t0 = 0 and not semisimple monodromy . . . . . . . . . . . . . . . . . . 47
7 Marked regular singular rank 2 (TE)-structures 51
8 Unfoldings of rank 2 (TE)-structures of type (Log) over a point 58
8.1 Classification results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8.2 (TE)-structures over given F -manifolds with Euler fields . . . . . . . . . . . . . . 63
8.3 Proof of Theorem 8.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
9 A family of rank 3 (TE)-structures with a functional parameter 70
References 72
1 Introduction
A holomorphic vector bundle H on C ×M , M a complex manifold, with a meromorphic con-
nection ∇ with a pole of Poincaré rank 1 along {0} × M and no pole elsewhere, is called
a (TE)-structure. The aim of this paper is the local classification of all rank 2 (TE)-structures,
over arbitrary germs
(
M, t0
)
of manifolds.
Before we talk about the results, we will put these structures into a context, motivate their
definition, mention their occurence in algebraic geometry, and formulate interesting problems.
The rank 2 case is the first interesting case and already very rich. In many aspects it is probably
typical for arbitrary rank, in some not. And it is certainly the only case where such a thorough
classification is feasible.
The pole of Poincaré rank 1 along {0} ×M of the pair (H,∇) means the following. Let t =
(t1, . . . , tn) be holomorphic coordinates on M with coordinate vector fields ∂1, . . . , ∂n, and let z
be the standard coordinate on C. Then ∇∂zσ for a holomorphic section σ ∈ O(H) of H is
in z−2O(H), and ∇∂jσ is in z−1O(H). The pole of order two along ∂z is the first case beyond
the easy and tame case of a pole of order 1, i.e., a logarithmic pole. The pole of order 1 along ∂i
gives a good variation property, a generalization of Griffiths transversality for variations of Hodge
structures. It is the most natural constraint for an isomonodromic family of bundles on C with
poles of order 2 at 0. So, a pole of Poincaré rank 1 is in some sense the first case beyond the case
of connections with logarithmic poles. (A pole of Poincaré rank r ∈ N0 is defined for example
in [21, Section 0.14].)
In algebraic geometry, such connections arise naturally. A distinguished case is the Fourier–
Laplace transformation (with respect to the coordinate z) of the Gauss–Manin connection of
a family of holomorphic functions with isolated singularities (see [10, Chapter 8] and [21, Chap-
ter VII]). The paper [10] defines (TERP )-structures, which are (TE)-structures with additional
real structure and pairing and which generalize variations of Hodge structures. Also the no-
tion (TEZP )-structure makes sense, which is a (TE)-structures with a flat Z-lattice bundle on
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 3
C∗×M and a certain pairing. A family of holomorphic functions with isolated singularities (and
some topological well-behavedness) gives rise to a (TEZP )-structure over the base space of the
family (see [9, Chapter 11.4] and [10, Chapter 8]).
In [9] and other papers of the author, a Torelli problem is considered. We formulate it
here as the following question: Does the (TEZP )-structure of a holomorphic function germ
with an isolated singularity determine the (TEZP )-structure of the universal unfolding of the
function germ? The first one is a (TE)-structure over a point t0. The second one is a (TE)-
structure over a germ
(
M, t0
)
of a manifold M . It it an unfolding of the first (TE)-structure
with a primitive Higgs field. The base space M is an F -manifold with Euler field.
We explain these notions. A second (TE)-structure over a manifold M is an unfolding of
a first (TE)-structure over a submanifold of M if the restriction of the second (TE)-structure
to the submanifold is isomorphic to the first (TE)-structure. If φ : M ′ →M is a morphism and
if (H,∇) is a (TE)-structure over M , then the pull back φ∗(H,∇) is a (TE)-structure over M ′.
An unfolding of a (TE)-structure is universal if it induces any unfolding via a unique map φ
(see Definition 3.15(b)+(c) for details).
If (H → C × M,∇) is a (TE)-structure, then define the vector bundle K := H|{0}×M
on M and the Higgs field C := [z∇] ∈ Ω1(M,End(K)) on K. The endomorpisms CX =
[z∇X ] : O(K) → O(K) for X ∈ TM commute with one another, and they commute with the
endomorphism U :=
[
z2∇∂z
]
: O(K) → O(K) (see Definition 3.8 and Lemma 3.12). The Higgs
field C is primitive if on each sufficiently small subset U ⊂ K a section ζU exists such that the
map TU → O(K), X 7→ CXζU , is an isomorphism (see Definition 3.13).
An F -manifold with Euler field is a complex manifold M together with a holomorphic com-
mutative and associative multiplication ◦ on TM which comes equipped with the integrability
condition (2.1), with a unit field e ∈ TM (with ◦e = id) and an Euler field E ∈ TM with
LieE(◦) = ◦ (see [12] or Definition 2.1). A (TE)-structure over M with primitive Higgs field
induces on the base manifold M the structure of an F -manifold with Euler field (see Theo-
rem 3.14 for details).
A result of Malgrange [18] (cited in Theorem 3.16(c)) says that a (TE)-structure over
a point t0 has a universal unfolding if the endomorphism U : K → K (here K is a vector
space) is regular, i.e., it has only one Jordan block for each eigenvalue. Theorem 3.16(b) gives
a generalization from [13]. A special case of this generalization says that a (TE)-structure with
primitive Higgs field over a germ
(
M, t0
)
is its own universal unfolding (see Theorem 3.16(a)).
A supplement from [4] says that then the base space is a regular F -manifold (see Definition 2.4
and Theorem 2.5).
Malgrange’s result gives a universal unfolding if one starts with a (TE)-structure over a point
whose endomorphism U is regular. However, if one starts with a (TE)-structure over a point
such that U is not regular, then in general it has no universal unfolding, and the study of all
its unfoldings becomes very interesting. The second half of this paper (Sections 6–8) studies
this situation in rank 2. The Torelli problem for a holomorphic function germ with an isolated
singularity is similar: The endomorphism U of its (TEZP )-structure is never regular (except if
the function has an A1-singularity), but I hope that the (TEZP )-structure determines neverthe-
less somehow the specific unfolding with primitive Higgs field, which comes from the universal
unfolding of the original function germ.
Now sufficient background is given. We describe the contents of this paper.
The short Section 2 recalls the classification of the 2-dimensional germs of F -manifolds with
Euler fields (Theorem 2.2 from [9] and Theorem 2.3 from [6]). It treats also regular F -manifolds
(Definition 2.4 and Theorem 2.5 from [4]).
Section 3 recalls many general facts on (TE)-structures: their definition, their presentation by
matrices, formal (TE)-structures, unfoldings and universal unfoldings of (TE)-structures, Mal-
grange’s result and the generalization in [13], (TE)-structures over F -manifolds, (TE)-structures
4 C. Hertling
with primitive Higgs fields, regular singular (TE)-structures and elementary sections, Birkhoff
normal form for (TE)-structures (not all have one, Theorem 3.20 cites existence results of Ple-
mely and of Bolibroukh and Kostov). Not written before, but elementary is a correspondence
between (TE)-structures with trace free endomorphism U and arbitrary (TE)-structures (Lem-
mata 3.9, 3.10 and 3.11).
New is the notion of a marked (TE)-structure. It is needed for the construction of moduli
spaces. Theorem 3.29 (which builds on results in [15]) constructs such moduli spaces, but only
in the case of regular singular (TE)-structures. It starts with a good family of regular singular
(TE)-structures. There are two open problems. It is not clear how to generalize this notion of
a good family beyond the case of regular singular (TE)-structures. We hope, but did not prove
for rank ≥ 3, that any regular singular (TE)-structure (over M with dimM ≥ 1) is a good
family of regular singular (TE)-structures. For rank 2 this is true, it follows from Theorem 8.5.
Section 4 gives the classification of rank 2 (TE)-structures over a point t0. There are 4 types,
which we call (Sem), (Bra), (Reg) and (Log) (for semisimple, branched, regular singular and
logarithmic). In the type (Sem) U has two different eigenvalues, in the type (Log) U ∈ C · id,
in the types (Bra) and (Reg) U has a 2 × 2 Jordan block. In the cases when U is trace free,
a (TE)-structure of type (Log) has a logarithmic pole, a (TE)-structure of type (Reg) has
a regular singular, but not logarithmic pole, and the pull back of a (TE)-structure of type (Bra)
by a branched cover of C of order 4 has a meromorphic connection with semisimple pole of order 3
(see Lemma 4.9). The semisimple case (Sem) is not central in this paper. Therefore we do not
discuss it in detail and do not introduce Stokes structures. For the other types (Bra), (Reg)
and (Log), Section 4 discusses normal forms and their parameters. All (TE)-structures of type
(Bra) have nice Birkhoff normal forms (Theorem 4.11), but not all of type (Reg) (Theorem 4.17
and Remark 4.19) and type (Log) (Theorem 4.20 and Remark 4.22). The types (Reg) and (Log)
become transparent by the use of elementary sections.
A (TE)-structure of type (Sem) or (Bra) or (Reg) over a point t0 satisfies the hypothesis of
Malgrange’s result, namely, the endomorphism U : K → K is regular. Therefore it has a univer-
sal unfolding, and any unfolding of it is induced by this universal unfolding. Section 5 discusses
this. Also because of this fact, the semisimple case is not central in this paper.
Sections 6–8 are devoted to the study of (TE)-structures over a germ
(
M, t0
)
such that
the restriction to t0 is a (TE)-structure of type (Log). Then the set of points over which the
(TE)-structure restricts to one of type (Log) is either a hypersurface or the whole of M . In the
first case, it restricts to a fixed generic type (Sem) or (Bra) or (Reg) over points not in the
hypersurface. In the second case, the generic type is (Log).
Section 6 starts this study. It considers the cases with trace free U and dimM = 1. It has three
parts. In the first part, invariants of such 1-parameter families are studied. In a surprisingly
direct way, constraints on the difference of the leading exponents (defined in Theorem 4.20)
of the logarithmic (TE)-structure over t0 are found, and the monodromy in the generic cases
(Sem) and (Bra) turns out to be semisimple (Theorem 6.2). By Plemely’s result (and our
direct calculations), these cases come equipped with Birkhoff normal forms. Theorem 6.3 in the
second part classifies all (TE)-structures over
(
M, t0
)
with trace free U , dimM = 1, logarithmic
restriction to t0 and Birkhoff normal form. Theorem 6.7 in the third part classifies all generically
regular singular (TE)-structures over
(
M, t0
)
with dimM = 1, logarithmic restriction to t0, and
whose monodromy has a 2×2 Jordan block. The majority of these cases has no Birkhoff normal
form. Theorems 6.3 and 6.7 overlap in the cases which have Birkhoff normal forms.
Section 7 makes the moduli spaces of marked regular singular (TE)-structures from Theo-
rem 3.28 explicit in the rank 2 cases. It builds on the classification results for the types (Reg)
and (Log) in Section 4. The long Theorem 7.4 describes the moduli spaces and offers 5 figures
in order to make this more transparent. The moduli spaces have countably many topological
components, and each component consists of an infinite chain of projective spaces which are
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 5
either the projective line P1 or the Hirzebruch surface F2 or F̃2 (which is obtained by blowing
down in F2 the unique (−2)-curve). These moduli spaces simplify in the generic case (Reg) the
main proof in Section 8, the proof of Theorem 8.5.
Section 8 gives complete classification results, from different points of view. It has three
parts. Theorem 8.1 lists all rank 2 (TE)-structures over a 2-dimensional germ
(
M, t0
)
such
that the restriction to t0 has a logarithmic pole, such that the Higgs field is generically primi-
tive, and such that the induced structure of an F -manifold with Euler field extends to all
of M . Theorem 8.1(d) offers explicit normal forms. Corollary 8.3 starts with any logarithmic
rank 2 (TE)-structure over a point t0 and lists the (TE)-structures in Theorem 8.1(d) which
unfold it.
Theorem 8.5 is the most fundamental result of Section 8. Table (8.4) in it is a sublist of the
(TE)-structures in Theorem 8.1(d). Theorem 8.5 states that any unfolding of a rank 2 (TE)-
structure of type (Log) over a point is induced by one (TE)-structure in table (8.4). In the generic
cases (Reg) and (Log) these are precisely those in Theorem 8.1(d) with primitive Higgs field,
but in the generic cases (Sem) and (Bra) table (8.4) contains many (TE)-structures with only
generically primitive Higgs field. All the (TE)-structures in table (8.4) are universal unfoldings
of themselves, also those with only generically primitive Higgs field. Almost all logarithmic
(TE)-structures over a point have several unfoldings which do not induce one another. Only the
logarithmic (TE)-structures over a point whose monodromy has a 2×2 Jordan block and whose
two leading exponents coincide have a universal unfolding. This follows from Theorem 8.5 and
Corollary 8.3.
The second part of Section 8 starts from the 2-dimensional F -manifolds with Euler fields and
discusses how many and which (TE)-structures exist over each of them. It turns out that the
nilpotent F -manifold N2 with the Euler field E = t1∂1 + tr2
(
1 + c3t
r−1
2
)
∂2 for r ≥ 2 (case (2.7)
in Theorem 2.3) does not have any (TE)-structure over it if c3 ̸= 0, and it has no (TE)-structure
with primitive Higgs field over it if c3 ̸= 0 or r ≥ 3. However, most 2-dimensional F -manifolds
with Euler fields have one or countably many families of (TE)-structures with 1 or 2 parameters
over them.
The third part of Section 8 is the proof of Theorem 8.5.
In many aspects, the (TE)-structures of rank 2 are probably typical also for higher rank.
But Section 9 makes one phenomenon explicit which arises only in rank ≥ 3. Section 9 presents
a family of rank 3 (TE)-structures with primitive Higgs fields over a fixed 3-dimensional globally
irreducible F -manifold with nowhere regular Euler field, such that the family has a functional
parameter. The example is essentially due to M. Saito, it is a Fourier–Laplace transformation
of the main example in a preliminary version of [23] (though he considers only the bundle and
connection over a 2-dimensional submanifold of the F -manifold).
This paper has some overlap with [6] and [7]. In [6] (TE)-structures over the 2-dimensional
F -manifold N2 (with all possible Euler fields) were studied. They are of generic types (Bra),
(Reg) or (Log). In [7, Chapter 8] (TE)-structures over the 2-dimensional F -manifolds I2(m)
were studied. They are of generic type (Sem). However, in [6] and [7] the focus was on (TE)-
structures with primitive Higgs fields. Those with generically primitive, but not primitive Higgs
fields were not considered. And the approach to the classification was very different. It relied
on the formal classification of rank 2 (T )-structures in [5]. The approach here is independent of
these three papers.
2 The two-dimensional F -manifolds and their Euler fields
F -manifolds were first defined in [12]. Their basic properties were developed in [9]. An overview
on them and on more recent results is given in [7].
6 C. Hertling
Definition 2.1.
(a) An F -manifold (M, ◦, e) (without Euler field) is a complex manifoldM with a holomorphic
commutative and associative multiplication ◦ on the holomorphic tangent bundle TM , and
with a global holomorphic vector field e ∈ TM with e◦ = id (e is called a unit field), which
satisfies the following integrability condition:
LieX◦Y (◦) = X ◦ LieY (◦) + Y ◦ LieX(◦) for X,Y ∈ TM . (2.1)
(b) Given an F -manifold (M, ◦, e), an Euler field on it is a global vector field E ∈ TM with
LieE(◦) = ◦.
In this paper we are mainly interested in the 2-dimensional F -manifolds and their Euler fields.
They were classified in [9].
Theorem 2.2 ([9, Theorem 4.7]). In dimension 2, (up to isomorphism) the germs of F -
manifolds fall into three types:
(a) The semisimple germ. It is called A2
1, and it can be given as follows
(M, 0) =
(
C2, 0
)
with coordinates u = (u1, u2) and ek =
∂
∂uk
,
e = e1 + e2, ej ◦ ek = δjk · ej .
Any Euler field takes the shape
E = (u1 + c1)e1 + (u2 + c2)e2 for some c1, c2 ∈ C. (2.2)
(b) Irreducible germs, which (i.e., some holomorphic representatives of them) are at generic
points semisimple. They form a series I2(m), m ∈ Z≥3. The germ of type I2(m) can be
given as follows
(M, 0) =
(
C2, 0
)
with coordinates t = (t1, t2) and ∂k :=
∂
∂tk
,
e = ∂1, ∂2 ◦ ∂2 = tm−2
2 e.
Any Euler field takes the shape
E = (t1 + c1)∂1 +
2
m
t2∂2 for some c1 ∈ C.
(c) An irreducible germ, such that the multiplication is everywhere irreducible. It is called N2,
and it can be given as follows
(M, 0) =
(
C2, 0
)
with coordinates t = (t1, t2) and ∂k :=
∂
∂tk
,
e = ∂1, ∂2 ◦ ∂2 = 0.
Any Euler field takes the shape
E = (t1 + c1)∂1 + g(t2)∂2 for some c1 ∈ C
and some function g(t2) ∈ C{t2}. (2.3)
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 7
The family of Euler fields in (2.3) on N2 can be reduced by coordinate changes, which
respect the multiplication of N2, to a family with two continuous parameters and one discrete
parameter. This classification is proved in [6]. It is recalled in Theorem 2.3. The group Aut(N2)
of automorphisms of the germ N2 of an F -manifold is the group of coordinate changes of
(
C2, 0
)
which respect the multiplication of N2. It is
Aut(N2) = {(t1, t2) 7→ (t1, λ(t2)) |λ ∈ C{t2} with λ′(0) ̸= 0 and λ(0) = 0}.
Theorem 2.3. Any Euler field on the germ N2 of an F -manifold can be brought by a coordinate
change in Aut(N2) to a unique one in the following family of Euler fields
E = (t1 + c1)∂1 + ∂2, (2.4)
E = (t1 + c1)∂1, (2.5)
E = (t1 + c1)∂1 + c2t2∂2, (2.6)
E = (t1 + c1)∂1 + tr2
(
1 + c3t
r−1
2
)
∂2, (2.7)
where c1, c3 ∈ C, c2 ∈ C∗ and r ∈ Z≥2. The group Aut(N2, E) of coordinate changes of
(
C2, 0
)
which respect the multiplication of N2 and this Euler field is
Aut(N2, E) = {(t1, t2) 7→ (t1, γ(t2)t2) | γ as in (2.8)},
Case (2.4) (2.5) (2.6) (2.7)
γ ∈ {1} C{t2}∗ C∗ {
e2πil/(r−1) | l ∈ Z
} (2.8)
A special class of F -manifolds, the regular F -manifolds, is related to a result of Malgrange
on universal unfoldings of (TE)-structures, see Remarks 3.17.
Definition 2.4 ([4, Definition 1.2]). A regular F -manifold is an F -manifold (M, ◦, e) with
Euler field E such that at each t ∈ M the endomorphism E ◦ |t : TtM → TtM is a regular
endomorphism, i.e., it has for each eigenvalue only one Jordan block.
Theorem 2.5 ([4, Theorem 1.3(ii)]). For each regular endomorphism of a finite dimensional C-
vector space, there is a unique (up to unique isomorphism) germ
(
M, t0
)
of a regular F -manifold
such that E ◦ |t0 is isomorphic to this endomorphism.
Remarks 2.6.
(i) For a normal form of this germ of an F -manifold, see [4, Theorem 1.3(i)].
(ii) In dimension 2, this theorem is an easy consequence of Theorems 2.2 and 2.3. The germs
of regular 2-dimensional F -manifolds are as follows:
(a) The germ A2
1 in Theorem 2.2(a) with any Euler field E = (u1 + c1)e1 + (u2 + c2)e2
as in (2.2) with c1, c2 ∈ C, c1 ̸= c2.
(b) The germ N2 in Theorem 2.2(c) with any Euler field E = (t1 + c1)∂1 + ∂2 as in (2.4)
with c1 ∈ C.
3 (TE)-structures in general
3.1 Definitions
A (TE)-structure is a holomorphic vector bundle on C × M , M a complex manifold, with
a meromorphic connection∇ with a pole of Poincaré rank 1 along {0}×M and no pole elsewhere.
Here we consider them together with the weaker notion of (T )-structure and the more rigid
notions of a (TL)-structure and a (TLE)-structure. The structures had been considered before
in [13], and they are related to structures in [21, Chapter VII] and in [20].
8 C. Hertling
Definition 3.1.
(a) Definition of a (T )-structure (H → C × M,∇): H → C × M is a holomorphic vector
bundle. ∇ is a map
∇ : O(H) → z−1OC×M · Ω1
M ⊗O(H), (3.1)
which satisfies the Leibniz rule,
∇X(a · s) = X(a) · s+ a · ∇Xs for X ∈ TM , a ∈ OC×M , s ∈ O(H),
and which is flat (with respect to X ∈ TM , not with respect to ∂z),
∇X∇Y −∇Y ∇X = ∇[X,Y ] for X,Y ∈ TM .
Equivalent: For any z ∈ C∗, the restriction of ∇ to H|{z}×M is a flat holomorphic connec-
tion.
(b) Definition of a (TE)-structure (H → C ×M,∇) : H → C ×M is a holomorphic vector
bundle. ∇ is a flat connection on H|C∗×M with a pole of Poincaré rank 1 along {0} ×M ,
so it is a map
∇ : O(H) →
(
z−1OC×M · Ω1
M + z−2OC×M · dz
)
⊗O(H)
which satisfies the Leibniz rule and is flat.
(c) Definition of a (TL)-structure
(
H → P1 ×M,∇
)
: H → P1 ×M is a holomorphic vector
bundle. ∇ is a map
∇ : O(H) →
(
z−1OP1×M +OP1×M
)
· Ω1
M ⊗O(H),
such that for any z ∈ P1 \ {0}, the restriction of ∇ to H|{z}×M is a flat connection. It is
called pure if for any t ∈M the restriction H|P1×{t} is a trivial holomorphic bundle on P1.
(d) Definition of a (TLE)-structure
(
H → P1 ×M,∇
)
: It is simultaneously a (TE)-structure
and a (TL)-structure, where the connection ∇ has a logarithmic pole along {∞} ×M .
The (TLE)-structure is called pure if the (TL)-structure is pure.
Remark 3.2. Here we write the data in Definition 3.1(a)–(b) and the compatibility conditions
between them in terms of matrices. Consider a (TE)-structure (H → C × M,∇) of rank
rkH = r ∈ N. We will fix the notations for a trivialization of the bundle H|U×M for some small
neighborhood U ⊂ C of 0. Trivialization means the choice of a basis v = (v1, . . . , vr) of the
bundle H|U×M . Also, we choose local coordinates t = (t1, . . . , tn) with coordinate vector fields
∂i = ∂/∂ti on M . We write
∇v = v · Ω with Ω =
r∑
i=1
z−1 ·Ai(z, t)dti + z−2B(z, t)dz, (3.2)
Ai(z, t) =
∑
k≥0
A
(k)
i zk ∈Mr×r(OU×M ), (3.3)
B(z, t) =
∑
k≥0
B(k)zk ∈Mr×r(OU×M ), (3.4)
with A
(k)
i , B(k) ∈ Mr×r(OM ), but this dependence on t ∈ M is usually not written explicity.
The flatness 0 = dΩ + Ω ∧ Ω of the connection ∇ says for i, j ∈ {1, . . . , n} with i ̸= j
0 = z∂iAj − z∂jAi + [Ai, Aj ], (3.5)
0 = z∂iB − z2∂zAi + zAi + [Ai, B]. (3.6)
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 9
These equations split into the parts for the different powers zk for k ≥ 0 as follows
(
with
A
(−1)
i = B(−1) = 0
)
,
0 = ∂iA
(k−1)
j − ∂jA
(k−1)
i +
k∑
l=0
[
A
(l)
i , A
(k−l)
j
]
, (3.7)
0 = ∂iB
(k−1) − (k − 2)A
(k−1)
i +
k∑
l=0
[
A
(l)
i , B
(k−l)
]
. (3.8)
In the case of a (T )-structure, B and all equations except (3.2) which contain B are dropped.
Consider a second (TE)-structure
(
H̃ → C×M, ∇̃
)
of rank r overM , where all data exceptM
are written with a tilde. Let v and ṽ be trivializations. A holomorphic isomorphism from
the first to the second (TE)-structure maps v · T to ṽ, where T = T (z, t) =
∑
k≥0 T
(k)zk ∈
Mr×r(O(C,0)×M ) with T (k) ∈Mr×r(OM ) and T (0) invertible satisfies
v · Ω · T + v · dT = ∇(v · T ) = v · T · Ω̃. (3.9)
Equation (3.9) says more explicitly
0 = z∂iT +Ai · T − T · Ãi, (3.10)
0 = z2∂zT +B · T − T · B̃. (3.11)
These equations split into the parts for the different powers zk for k ≥ 0 as follows (with
T (−1) := 0):
0 = ∂iT
(k−1) +
k∑
l=0
(
A
(l)
i · T (k−l) − T (k−l) · Ã(l)
i
)
,
0 = (k − 1)T (k−1) +
k∑
l=0
(
B(l) · T (k−l) − T (k−l) · B̃(l)
)
.
The isomorphism here fixes the base manifold M . Such isomorphisms are called gauge iso-
morphisms. A general isomorphism is a composition of a gauge isomorphism and a coordinate
change onM (a coordinate change induces an isomorphism of (TE)-structures, see Lemma 3.6).
Remark 3.3. In this paper we care mainly about (TE)-structures over the 2-dimensional germs
of F -manifolds with Euler fields. For each of them except (N2, E = (t1 + c1)∂1), the group of
coordinate changes of (M, 0) =
(
C2, 0
)
which respect the multiplication and E is quite small,
see Theorem 2.3. Therefore in this paper, we care mainly about gauge isomorphisms of the
(TE)-structures over these F -manifolds with Euler fields.
Definition 3.4. Let M be a complex manifold.
(a) The sheaf OM [[z]] onM is defined by OM [[z]](U) := OM (U)[[z]] for an open subset U ⊂M
(with OM (U) and OM [[z]](U) the sections of OM and OM [[z]] on U). Observe that the
germ (OM [[z]])t0 for t0 ∈ M consists of formal power series
∑
k≥0 fkz
k whose coefficients
fk ∈ OM,t0 have a common convergence domain. In the case of
(
M, t0
)
=
(
Cn, 0
)
we write
OCn [[z]]0 =: C{t, z]].
(b) A formal (T )-structure over M is a free OM [[z]]-module O(H) of some finite rank r ∈ N
together with a map ∇ as in (3.1), where OC×M is replaced by OM [[z]] which satisfies prop-
erties analogous to ∇ in Definition 3.1(a), i.e., the Leibniz rule for X ∈ TM , a ∈ OM [[z]],
s ∈ O(H) and the flatness condition for X,Y ∈ TM .
A formal (TE)-structure is defined analogously: In Definition 3.1(b) one has to replace
OC×M by OM [[z]].
10 C. Hertling
Remark 3.5. The formulas in Remark 3.2 hold also for formal (T )-structures and formal (TE)-
structures if one replaces OC×M , OU×M and O(C,0)×M by OM [[z]].
The following lemma is obvious.
Lemma 3.6. Let (H → C×M,∇) be a (TE)-structure over M , and let φ : M ′ →M be a holo-
morphic map between manifolds. One can pull back H and ∇ with id×φ : C ×M ′ → C ×M .
We call the pull back φ∗(H,∇). It is a (TE)-structure over M ′. We say that the pull back
φ∗(H,∇) is induced by the (TE)-structure (H,∇) via the map φ.
Remarks 3.7.
(i) We will give in Theorem 8.5 and in Corollary 5.1 and Lemma 5.2(iv) a classification of
rank 2 (TE)-structures over germs
(
M, t0
)
=
(
C2, 0
)
of 2-dimensional manifolds such that
any rank 2 (TE)-structure over a germ (M ′, s0) is obtained as the pull back φ∗(H,∇) of
a rank 2 (TE)-structure in the classification via a holomorphic map φ :
(
M ′, s0
)
→
(
M, t0
)
.
(ii) Here the behaviour of the (TE)-structure (H,∇) over
(
M, t0
)
=
(
C2, 0
)
with coordinates
t = (t1, t2) along t1 is quite trivial. It is convenient to split it off. The next subsection
does this in greater generality.
3.2 (TE)-structures with trace free pole part
Definition 3.8. Let (H → C ×M,∇) be a (TE)-structure. Define the vector bundle K :=
H|{0}×M over M . The pole part of the (TE)-structure is the endomorphism U : K → K which
is defined by
U :=
[
z2∇∂z
]
: K → K. (3.12)
The pole part is trace free if trU = 0 on M .
The following lemma gives formal invariants of a (TE)-structure.
Lemma 3.9. Let (H → C ×M,∇) be a (TE)-structure of rank r ∈ N over a manifold M .
By a formal invariant of the (TE)-structure, we mean an invariant of its formal isomorphism
class.
(a) Its pole part U , that means the pair (K,U) up to isomorphism, is a formal invariant
of the (TE)-structure. Especially, the holomorphic functions δ(0) := detU ∈ OM and
ρ(0) := 1
r trU ∈ OM are formal invariants.
(b) For any t0 ∈ M , fix an OM,t0-basis v of O(H)(0,t0), consider the matrices in (3.2)–(3.4),
consider the function ρ(1) := 1
r trB
(1) ∈ OM,t0, and consider the functions δ(k) ∈ OM,t0
for k ∈ N0 which are defined by writing detB as a power series
detB =
∑
k≥0
δ(k)zk.
Then the functions δ(1) and ρ(1) are independent of the choice of the basis v. The locally
for any t0 defined functions δ(1) and ρ(1) glue to global holomorphic functions δ(1) ∈ OM
and ρ(1) ∈ OM . They are formal invariants. Furthermore, the function ρ(1) is constant on
any component of M .
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 11
Proof. U , δ(0), ρ(0) and δ(1) are formal invariants because of (3.11): B̃ = T−1BT +z2 ·T−1∂zT.
For ρ(1), observe additionally
B̃(1) =
(
T (0)
)−1
B(1)T (0) +
[(
T (0)
)−1
B(0)T (0),
(
T (0)
)−1
T (1)
]
.
Recall also that the trace of a commutator of matrices is 0. Therefore ρ(1) is a formal invariant.
Equation (3.8) for k = 2 implies ∂i tr
(
B(1)
)
= 0, so the function ρ(1) is constant. ■
The following lemma is obvious.
Lemma 3.10. Let (H → C×M,∇) be a (TE)-structure of rank r ∈ N over a manifold M .
(a) Consider a holomorphic function g : M → C. The trivial line bundle H [1] = C×(C×M) →
C ×M over C ×M with connection ∇[1] := d + d
(g
z
)
defines a (TE)-structure of rank 1
over M , whose sheaf of sections with connection is called Eg/z.
(b) (O(H),∇)⊗ Eg/z for g as in (a) is a (TE)-structure.
(c) The (TE)-structure
(
H [2] → C×M,∇[2]
)
with
(
O
(
H [2]
)
,∇[2]
)
= (O(H),∇)⊗ Eρ(0)/z has
trace free pole part. And, of course, (O(H),∇) ∼=
(
O
(
H [2]
)
,∇[2]
)
⊗ E−ρ(0)/z. If v is a
C{t, z}-basis of O(H)0 = O
(
H [2]
)
0
, then the matrix valued connection 1-forms Ω and Ω[2]
of ∇ and ∇[2] with respect to this basis satisfy Ω = Ω[2] − d
(ρ(0)
z
)
· 1r.
(d) (Definition) Consider a (TE)-structure
(
H [3] → C ×M [3],∇[3]
)
with trace free pole part.
Consider the manifold M [4] := C ×M [3] with (local) coordinates t1 on C and t′ on M [3],
and the projection φ[4] : M [4] → M [3], (t1, t
′) 7→ t′. Define the (TE)-structure
(
H [4] →
C×M [4],∇[4]
)
with
(
O
(
H [4]
)
,∇[4]
)
=
(
φ[4]
)∗(O(H [3]
)
,∇[3]
)
⊗ E t1/z.
(e) If the (TE)-structure
(
H [2],∇[2]
)
is induced by the (TE)-structure
(
H [3],∇[3]
)
via a map
φ : M →M [3], then the (TE)-structure (H,∇) is induced by the (TE)-structure
(
H [4],∇[4]
)
via the map (−ρ(0), φ) : M →M [4] = C×M [3].
Part (c) allows to go from an arbitrary (TE)-structure to one with trace free pole part, and
to go back to the original one. Part (e) considers two (TE)-structures as in part (c), an original
one and an associated one with trace free pole part. If the associated one is induced by a third
(TE)-structure, then the original one is induced by a closely related (TE)-structure with one
parameter more. Lemma 3.11 continues Lemma 3.10.
Lemma 3.11. Let
(
H → C ×
(
M, t0
)
,∇
)
be a (TE)-structure of rank r ∈ N over a germ(
M, t0
)
of a manifold, with coordinates t = (t1, . . . , tn) and ∂i := ∂/∂ti. We suppose t0 = 0
so that O(C×M,(0,t0)) = C{t, z}. Recall the functions ρ(0) and ρ(1) of the (TE)-structure from
Lemma 3.9.
Consider the (TE)-structure
(
H [2],∇[2]
)
from Lemma 3.10 with trace free pole part which is
defined by
(
O
(
H [2]
)
,∇[2]
)
:= (O(H),∇) ⊗ Eρ(0)/z. Here H [2] = H, but ∇[2] = ∇ + d
(ρ(0)
z
)
· id.
The matrices Ai and B in (3.2)–(3.4) for the (TE)-structure
(
H [2],∇[2]
)
of any C{t, z}-basis v
of O
(
H [2]
)
0
satisfy
0 = trA
(0)
i = trB(0) = tr
(
B(1) − ρ(1)1r
)
. (3.13)
The basis v can be chosen such that the matrices satisfy
0 = trAi = tr
(
B − zρ(1)1r
)
. (3.14)
12 C. Hertling
Proof. Any C{t, z}-basis v of O
(
H [2]
)
0
= O(H)0 satisfies
trB(0) = trU [2] = 0 as
(
H [2],∇[2]
)
has trace free pole part,
trA
(0)
i = 0 because of tr ∂iB
(0) = ∂i trB
(0) = 0 and (3.8) for k = 1,
tr
(
B(1) − ρ(1)1r
)
= 0 by Lemma 3.9 and especially,
Ω = Ω[2] − d
(
ρ(0)
z
)
· 1r = Ω[2] −
n∑
i=1
z−1∂ρ
(0)
∂ti
· 1rdti + z−2ρ(0) · 1rdz.
Start with an arbitrary basis v, consider the function
g :=
1
r
∑
k≥2
− trB(k)
k − 1
· zk−1 ∈ zC{t, z}, (3.15)
consider T := eg · 1r, and ṽ := v · T . (3.11) gives
B̃ = B + T−1z2∂zT = B +
(
−
∑
k≥2
trB(k)zk
)
· 1
r
1r,
so tr B̃(k) = 0 for k ≥ 2, B̃(1) = B(1), B̃(0) = B(0).
Therefore now suppose tr
(
B − zρ(1)1r
)
= 0. (3.8) for k ≥ 3 gives trA
(l)
i = 0 for l ≥ 2,
because tr ∂iB
(l) = ∂i trB
(l) = 0.
Finally, we consider T = T (0) = eh · 1r for a suitable function h ∈ C{t}. Then B̃ = B,
Ã
(k)
i = A
(k)
i for k ̸= 1, and Ã
(1)
i = A
(1)
i + ∂ih · 1r. So we need h ∈ C{t} with ∂ih = −1
r trA
(1)
i .
Such a function exists because (3.7) for k = 2 implies ∂i trA
(1)
j = ∂j trA
(1)
i . We have obtained
a basis v with tr
(
B − zρ(1)1r
)
= 0 and trAi = 0 for all i. ■
3.3 (TE)-structures over F -manifolds with Euler fields
The pole part of a (T )-structure (or a (TE)-structure) over C×M along {0}×M induces a Higgs
bundle (together with U). This is elementary (e.g., [5] or [10]).
Lemma 3.12. Let (H → C ×M,∇) be a (T )-structure. Define K := H|{0}×M . Then C :=
[z∇] ∈ Ω1(M,End(K)), more explicitly
CX [a] := [z∇Xa] for X ∈ TM , a ∈ O(H), (3.16)
is a Higgs field, i.e., the endomorphisms CX , CY : K → K for X,Y ∈ TM commute.
If (H → C × M) is a (TE)-structure, then its pole part U : K → K commutes with all
endomorphisms CX , X ∈ TM , short: [C,U ] = 0.
Definition 3.13. The Higgs field of a (T )-structure or a (TE)-structure (H → C ×M,∇) is
primitive if there is an open cover V ofM and for any U ∈ V a section ζU ∈ O(K|U ) (called a local
primitive section) with the property that the map TU ∋ X → CXζU ∈ O(K) is an isomorphism.
Theorems 3.14 and 3.16 show in two ways that primitivity of a Higgs field is a good condition.
Theorem 3.14 was first proved in [11, Theorem 3.3] (but see also [5, Lemma 10]).
Theorem 3.14. A (T )-structure (H → C ×M,∇) with primitive Higgs field induces a multi-
plication ◦ on TM which makes M an F -manifold. A (TE)-structure (H → C ×M,∇) with
primitive Higgs field induces in addition a vector field E on M , which, together with ◦, makes M
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 13
an F -manifold with Euler field. The multiplication ◦, unit field e and Euler field E (the latter
in the case of a (TE)-structure), are defined by
CX◦Y = CXCY , Ce = Id, CE = −U ,
where C is the Higgs field defined by ∇, and U is defined in (3.12).
Definition 3.15 recalls the notions of an unfolding and of a universal unfolding of a (TE)-
structure over a germ of a manifold from [13, Definition 2.3]. It turns out that any (TE)-
structure over a germ of a manifold with primitive Higgs field is a universal unfolding of itself.
Interestingly, we will see in Theorem 8.5 also (TE)-structures which are universal unfoldings
of themselves, but where the Higgs bundle is only generically primitive. Still in the examples
which we consider, the base manifold is an F -manifold with Euler field globally.
Malgrange [18] proved that a (TE)-structure over a point t0 has a universal unfolding with
primitive Higgs field if the endomorphism U : Kt0 → Kt0 is regular, i.e., it has for each eigenvalue
only one Jordan block. A generalization was given by Hertling and Manin [13, Theorem 2.5].
Theorem 3.16 cites in part (b) the generalization. Part (a) is the special case of a (TE)-
structure with primitive Higgs field. Part (c) is the special case of a (TE)-structure over a point,
Malgrange’s result.
Definition 3.15. Let
(
H → C×
(
M, t0
)
,∇
)
be a (TE)-structure over a germ
(
M, t0
)
of a mani-
fold.
(a) An unfolding of it is a (TE)-structure
(
H [1] → C ×
(
M × Cl1 ,
(
t0, 0
))
,∇[1]
)
over a germ(
M × Cl1 ,
(
t0, 0
))
(for some l1 ∈ N0) together with a fixed isomorphism
i[1] :
(
H → C×
(
M, t0
)
,∇
)
→
(
H [1] → C×
(
M × Cl1 ,
(
t0, 0
))
,∇[1]
)
|C×(M×{0},(t0,0)).
(b) One unfolding
(
H [1] → C×
(
M×Cl1 ,
(
t0, 0
))
,∇[1], i[1]
)
induces a second unfolding
(
H [2] →
C×
(
M × Cl2 ,
(
t0, 0
))
,∇[2], i[2]
)
if there are a holomorphic map germ
φ :
(
M × Cl2 ,
(
t0, 0
))
→
(
M × Cl1 ,
(
t0, 0
))
,
which is the identity on M ×{0}, and an isomorphism j from the second unfolding to the
pullback of the first unfolding by φ such that
i[1] = j|C×(M×{0},(t0,0)) ◦ i[2]. (3.17)
(Then j is uniquely determined by φ and (3.17).)
(c) An unfolding is universal if it induces any unfolding via a unique map φ.
By definition of a universal unfolding in part (c), a (TE)-structure has (up to canonical
isomorphism) at most one universal unfolding, because any two universal unfoldings induce one
another by unique maps.
Theorem 3.16.
(a) ([13, Theorem 2.5]) A (TE)-structure over a germ
(
M, t0
)
with primitive Higgs field is
a universal unfolding of itself.
(b) ([13, Theorem 2.5]) Let
(
H → C ×
(
M, t0
)
,∇
)
be a (TE)-structure over a germ
(
M, t0
)
of a manifold. Let
(
K →
(
M, t0
)
, C
)
be the induced Higgs bundle over
(
M, t0
)
. Suppose
that a vector ζt0 ∈ Kt0 with the following properties exists:
14 C. Hertling
(IC) (Injectivity condition) The map C•ζt0 : Tt0M → Kt0 is injective.
(GC) (Generation condition) ζt0 and its images under iteration of the maps U|t0 : Kt0 →
Kt0 and CX : Kt0 → Kt0 for X ∈ Tt0M generate Kt0.
Then a universal unfolding of the (TE)-structure over a germ
(
M × Cl,
(
t0, 0
))
(l ∈ N0
suitable) exists. It is unique up to isomorphism. Its Higgs field is primitive.
(c) ([18]) A (TE)-structure over a point t0 has a universal unfolding with primitive Higgs field
if the endomorphism
[
z2∇∂z
]
= U : Kt0 → Kt0 is regular, i.e., it has for each eigenvalue
only one Jordan block. In that case, the germ of the F -manifold with Euler field which
underlies the universal unfolding, is by definition (Definition 2.4) regular.
Remarks 3.17.
(i) As said above, the parts (a) and (c) are special cases of part (b).
(ii) A germ
((
M, t0
)
, ◦, e, E
)
of a regular F -manifold is uniquely determined by the regular
endomorphism E ◦ |t0 : Tt0 → Tt0 (Theorem 2.5).
(iii) Consider the germ (M, 0) =
(
C2, 0
)
of a 2-dimensional F -manifold with Euler field E
in Theorem 2.2. It is regular if and only if E ◦ |t=0 /∈ {λ id |λ ∈ C}. In the semisimple
case (Theorem 2.2(a)) this holds if and only if c1 ̸= c2. In the cases I2(m) (m ≥ 3) it does
not hold. In the case of N2 with E = t1∂1 + g(t2)∂2 it holds if and only if g(0) ̸= 0. See
also Remark 2.6(ii).
(iv) Theorem 3.16(c) implies that a (TE)-structure with primitive Higgs field over a germ(
M, t0
)
of a regular F -manifold with Euler field is determined up to gauge isomorphism
by the restriction of the (TE)-structure to t0.
(v) Lemma 3.6, Definition 3.8, Lemmata 3.9–3.12, Definition 3.13, Theorem 3.14 and Defini-
tion 3.15 hold or make sense also for formal (T )-structures or (TE)-structures. However,
the proof of Theorem 3.16 used in an essential way holomorphic (TE)-structures. We do
not know whether Theorem 3.16 holds also for formal (TE)-structures.
3.4 Birkhoff normal form
Definition 3.18. Let (H → C×M,∇) be a (TE)-structure over a manifoldM with coordinates
t = (t1, . . . , tn). A Birkhoff normal form consists of a basis v of H and associated matrices
A1, . . . , An, B as in (3.2) such that
A
(k)
1 = · · · = A(k)
n = 0 for k ≥ 1, B(k) = 0 for k ≥ 2, ∂iB
(1) = 0.
Remarks 3.19.
(i) Such a basis defines an extension of the (TE)-structure to a pure (TLE)-structure. Then
it is a basis of the (TLE)-structure whose restriction to {∞} ×M is flat with respect to
the residual connection (that is just the restriction of the connection ∇ of the underlying
(TL)-structure to H|{∞}×M ). Then the conditions (3.7) and (3.8) boil down to
0 =
[
A
(0)
i , A
(0)
j
]
, ∂iA
(0)
j = ∂jA
(0)
i , (3.18)
0 =
[
A
(0)
i , B(0)
]
, ∂iB
(0) +A
(0)
i +
[
A
(0)
i , B(1)
]
, 0 = ∂iB
(1). (3.19)
Such a basis is relevant for the construction of Frobenius manifolds (see, e.g., [7]).
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 15
(ii) Vice versa, if the (TE)-structure has an extension to a pure (TLE)-structure, then a basis v
of the (TLE)-structure exists whose restriction to {∞} ×M is flat with respect to the
residual connection. Then this basis v and the associated matrices form a Birkhoff normal
form.
(iii) A Birkhoff normal form does not always exist. But if a Birkhoff normal form of the
restriction of a (TE)-structure over M to a point t0 ∈ M exists, it extends to a Birkhoff
normal form of the (TE)-structure over the germ
(
M, t0
)
[21, Chapter VI, Theorem 2.1]
(or [7, Theorem 5.1(c)]).
(iv) The problem whether a (TE)-structure over a point has an extension to a pure (TLE)-
structure is a special case of the Birkhoff problem, which itself is a special case of the
Riemann–Hilbert–Birkhoff problem. The book [1] and Chapter IV in [21] are devoted to
these problems and results on them.
Here the following two results on the Birkhoff problem will be useful. However, we will use
part (a) only in the case of a (TE)-structure over a point t0 with a logarithmic pole at z = 0,
in which case it is trivial.
Theorem 3.20. Let
(
H → C×
{
t0
}
,∇
)
be a (TE)-structure over a point t0.
(a) (Plemely, [21, Chapter IV, Corollary 2.6(1)]) If the monodromy is semisimple, the (TE)-
structure has an extension to a pure (TLE)-structure.
(b) (Bolibroukh and Kostov, [21, Chapter IV, Corollary 2.6(3)]) The germ O(H)0 ⊗C{z}
C{z}
[
z−1
]
is a C{z}
[
z−1
]
-vector space of dimension r = rkH ∈ N on which ∇ acts.
If no C{z}
[
z−1
]
sub vector space of dimension in {1, . . . , r−1} exists which is ∇-invariant,
then the (TE)-structure has an extension to a pure (TLE)-structure.
3.5 Regular singular (TE)-structures
A (TE)-structure over a point t0 is regular singular if all its holomorphic sections have moderate
growth near 0. A good tool to treat this situation are special sections of moderate growth, the
elementary sections. Definition 3.21 explains them and other basic notations. We work with
a simply connected manifold M , so that the only monodromy is the monodromy along closed
paths in the punctured z-plane going around 0. One important case is the case of a germ
(
M, t0
)
of a manifold. The most important case is the case of a point, M =
{
t0
}
.
Definition 3.21. Let (H → C×M,∇) be a (TE)-structure of rank r = rkH ∈ N over a simply
connected manifold M . We associate the following data to it.
(a) H ′ := H|C∗×M is the flat bundle on C∗ ×M . H∞ denotes the C-vector space (of dimen-
sion r) of global flat multivalued sections on H ′. Let the endomorphism Mmon : H∞ →
H∞ be the monodromy on it with semisimple part Mmon
s , unipotent part Mmon
u (with
Mmon
s Mmon
u = Mmon
u Mmon
s ), nilpotent part Nmon := logMmon
u so that Mmon
u = eN
mon
,
and with eigenvalues in the finite set Eig(Mmon) ⊂ C. For λ ∈ C, let
H∞
λ := ker
(
Mmon
s − λ id : H∞ → H∞)
be the generalized eigenspace in H∞ of the monodromy with eigenvalue λ. It is not {0} if
and only if λ ∈ Eig(Mmon).
(b) For α ∈ C, define the finite dimensional C-vector space Cα of the following global sections
of H ′,
Cα :=
{
σ ∈ O(H ′)(C∗) | (∇z∂z − α id)r(σ) = 0,∇∂i(σ) = 0
}
16 C. Hertling
(where t = (t1, . . . , tn) are local coordinate and ∂i are the coordinate vector fields). Observe
zk · Cα = Cα+k for k ∈ Z. For each α the map
s(·, α) : H∞
e−2πiα → Cα,
A 7→ s(A,α) := zα · e− log z·Nmon/2πiA(log z),
is an isomorphism. So, Cα ̸= {0} if and only if e−2πiα ∈ Eig(Mmon). The sections s(A,α)
are called elementary sections.
(c) A holomorphic section σ of H ′|(U1\{0})×U2
for U1 ⊂ C a neighborhood of 0 ∈ C and U2 ⊂M
open in M can be written uniquely as an (in general infinite) sum of elementary sections
es(σ, α) ∈ OU2 · Cα with coefficients in OU2 ,
σ =
∑
α : e−2πiα∈Eig(Mmon)
es(σ, α).
In order to see this, choose numbers αj ∈ C and elementary sections sj ∈ Cαj for j ∈
{1, . . . , r} such that s1, . . . , sr form a global basis of H ′. Then
σ =
r∑
j=1
ajsj with aj = aj(z, t) =
∞∑
k=−∞
akj(t)z
k ∈ O(U1\{0})×U2
. (3.20)
Here (3.20) is the expansion of aj as a Laurent series in z with holomorphic coefficients
akj ∈ OU2 in t. Then
es(σ, α)(z, t) =
∑
j : α−αj∈Z
aα−αj ,j(t)z
α−αjsj .
(d) A holomorphic section σ as in (c) has moderate growth if a bound b ∈ R with es(σ, α) = 0
for all α with Re(α) < b exists. The sheaf V>−∞ on C ×M of all sections of moderate
growth is
V>−∞ :=
⊕
α : −1<Re(α)≤0
OC×M
[
z−1
]
· Cα.
The Kashiwara–Malgrange V -filtration is given by the locally free subsheaves for r ∈ R,
Vr :=
⊕
α : Re(α)∈[r,r+1[
OC×M · Cα.
Definition 3.22.
(a) A (TE)-structure (H → C×M,∇) over a simply connected manifoldM is regular singular
if O(H) ⊂ V>−∞, so if all its holomorphic sections have moderate growth near 0.
(b) A (TE)-structure (H → C×M,∇) over a simply connected manifoldM is logarithmic if it
has a basis v whose connection 1-form Ω has a logarithmic pole along {0} ×M (then this
holds for any basis). In the notations of (3.2)–(3.4) that means A
(0)
i = B(0) = 0. Then the
restriction of ∇ to K := H|{0}×M is well-defined. It is called the residual connection ∇res.
The residue endomorphism is Res0 = [∇z∂z ] : K → K.
Theorem 3.23 (well known, e.g., [9, Theorems 7.10 and 8.7]). Let (H → C × M,∇) with
H|C∗×M = H ′ be a logarithmic (TE)-structure over a simply connected manifold.
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 17
(a) The bundle H has a global basis which consists of elementary sections sj ∈ Cαj , j ∈
{1, . . . , rkH}, for some αj ∈ C. Especially, (O(H),∇) = φ∗
t0
(
O
(
H|C×{t0}
)
,∇
)
for any
t0 ∈M , where φt0 : M →
{
t0
}
is the projection. So it is just the pull back of a logarithmic
(TE)-structure over a point. Especially, it is a regular singular (TE)-structure.
(b) The residual connection ∇res is flat. In the notations (3.2)–(3.4), its connection 1-form is∑n
i=1A
(1)
i dti. The residue endomorphism Res is ∇res-flat. In the notations (3.2)–(3.4),
it is given by B(1).
(c) The endomorphism e−2πiRes0 : K → K has the same eigenvalues as the monodromy Mmon,
but it might have a simpler Jordan block structure. If no eigenvalues of Res0 differ by
a nonzero integer (nonresonance condition) then e−2πiRes0 has the same Jordan block struc-
ture as the monodromy Mmon.
Remarks 3.24.
(i) Part (a) of Theorem 3.23 implies that a logarithmic (TE)-structure over a simply connected
manifold M is the pull back φ∗((H,∇)|C×{t0}) of its restriction to t0 for any t0 ∈M .
(ii) In the case of a regular singular (TE)-structure over a simply connected manifold M ,
one can choose elementary sections sj ∈ Cαj , j ∈ {1, . . . , rkH}, for some αj ∈ C, such
that they form a basis of H∗ and such the extension to {0} ×M which they define, is
a logarithmic (TE)-structure. Then the base change from any local basis of H to the basis
(s1, . . . , srkH) of this new (TE)-structure is meromorphic, so the two (TE)-structures give
the same meromorphic bundle. This observation fits to the usual definition of meromorphic
bundle with regular singular pole.
(iii) The property of a section to have moderate growth, is invariant under pull back. Therefore
also the property of a (TE)-structure to be regular singular is invariant under pull back.
3.6 Marked (TE)-structures and moduli spaces for them
It is easy to give a (TE)-structure (H → C ×M,∇) with nontrivial Higgs field and which is
thus not the pull back of the (TE)-structure over a point, such that nevertheless the (TE)-
structures over all points t0 ∈ M are isomorphic as abstract (TE)-structures. Examples are
given in Remark 7.1(ii). The existence of such (TE)-structures obstructs the construction of nice
Hausdorff moduli spaces for (TE)-structures up to isomorphism. The notion of a marked (TE)-
structure hopefully remedies this. However, in the moment, we have only results in the regular
singular cases. Definition 3.25 gives the notion of a marked (TE)-structure. Definition 3.26
defines good families of marked regular singular (TE)-structures. Definition 3.28 defines a functor
for such families. Theorem 3.29 states that this functor is represented by a complex space.
It builds on results in [15, Chapter 7]. Several remarks discuss what is missing in the other cases
and what more we have in the regular singular rank 2 case, thanks to Theorems 6.3, 6.7 and 8.5.
Definition 3.25.
(a) A reference pair
(
Href,∞,M ref
)
consists of a finite dimensional (reference) C-vector space
Href,∞ together with an automorphism M ref of it.
(b) Let M be a simply connected manifold. A marking on a (TE)-structure (H → C×M,∇)
is an isomorphism ψ : (H∞,Mmon) →
(
Href,∞,M ref
)
. Here H∞ is (as in Definition 3.16)
the space of global flat multivalued sections on the flat bundle H ′ := H|C∗×M , and Mmon
is its monodromy.
(
Href,∞,M ref
)
is a reference pair. The isomorphism ψ of pairs means
an isomorphism ψ : H∞ → Href,∞ with ψ ◦Mmon = M ref ◦ ψ. A marked (TE)-structure
is a (TE)-structure with a marking.
18 C. Hertling
(c) An isomorphism between two marked (TE)-structures
((
H(1),∇(1)
)
, ψ(1)
)
and((
H(2),∇(2)
)
, ψ(2)
)
over the same base space M (1) = M (2) and with the same reference
pair
(
Href,∞,M ref
)
is a gauge isomorphism φ between the unmarked (TE)-structures such
that the induced isomorphism φ∞ : H(1),∞ → H(2),∞ is compatible with the marking,
ψ(2) ◦ φ∞ = ψ(1).
(d) Set(H
ref,∞,Mref) denotes the set of marked (TE)-structures over a point with the same
reference pair
(
Href,∞,M ref
)
. Furthermore, Set(H
ref,∞,Mref),reg ⊂ Set(H
ref,∞,Mref) denotes
the subset of marked regular singular (TE)-structures over a point with the same reference
pair
(
Href,∞,M ref
)
.
We hope that Set(H
ref,∞,Mref) carries for any reference pair
(
Href,∞,M ref
)
a natural structure
as a complex space. Theorem 3.29 says that this holds for Set(H
ref,∞,Mref),reg and that this space
represents a functor of good families of marked regular singular (TE)-structures. Definition 3.26
gives a notion of a family of marked (TE)-structures and the notion of a good family of marked
regular singular (TE)-structures.
Definition 3.26. Let X be a complex space. Let t0 be an abstract point and φ : X →
{
t0
}
be the projection. Let
(
Href,∞,M ref
)
be a reference pair. Let
(
Href,∗,∇ref
)
be a flat bundle
on C∗ ×
{
t0
}
with monodromy M ref and whose space of global flat multivalued sections is
identified with Href,∞. Let i : C∗ ×X ↪→ C×X be the inclusion.
(a) A family of marked (TE)-structures over X is a pair (H,ψ) with the following properties:
(i) H is a holomorphic vector bundle on C×X, i.e., the linear space associated to a locally
free sheaf O(H) of OC×X -modules. Denote H ′ := H|C∗×X .
(ii) ψ is an isomorphism ψ : H ′ → φ∗Href,∗ such that the restriction of the induced flat
connection on H ′ to C∗ × {x} for any x ∈ X makes H|C×{x} into a (TE)-structure
over the point x, i.e., the connection has a pole of order ≤ 2 on holomorphic sections
of H|C×{x}.
(b) Consider a family (H,ψ) of marked regular singular (TE)-structures over X. The mark-
ing ψ induces for each x ∈ X canonical isomorphisms
ψ : H∞(x) → Href,∞,
ψ : Cα(x) → Cref,α
(
α ∈ C with e−2πiα ∈ Eig
(
M ref
))
,
ψ : V r(x) → V ref,r (r ∈ R),
where H∞(x), Cα(x), V r(x) and Cref,α, V ref,r are defined for the (TE)-structure over x respec-
tively for (Href,∗,∇) as in Definition 3.21.
The family (H,ψ) is called good if some r ∈ R and some N ∈ N exist which satisfy
O(H|C×{x})0 ⊃ V r(x) for any x ∈ X, (3.21)
dimCO(H|C×{x})0/V
r(x) = N for any x ∈ X. (3.22)
Remarks 3.27.
(i) The notion of a family of marked (TE)-structures is too weak. For example, it contains the
following pathological family of logarithmic (TE)-structures of rank 1 over X := C (with
coordinate t) and with trivial monodromy. Write s0 ∈ C0 for a generating flat section.
Define H by
O(H) = OC×X ·
(
t+ zl
)
s0 for some l ∈ N.
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 19
The marked (TE)-structures over all points t ∈ C∗ ⊂ X = C are isomorphic and even
equal, the one over t = 0 is different. The dimension O
(
H|C×{t}
)
/V l(t) is equal to l
for t ∈ C∗ and equal to 0 for t = 0. Therefore this family is not good in the sense of
Definition 3.26(b). Also, z∇∂z
(
t+zl
)
s0 = lzls0 is not a section in O(H), although for each
fixed t ∈ X, the restriction to C× {t} is a section in O(H|C×{t}).
(ii) Theorem 3.29 gives evidence that the notion of a good family of marked regular singular
(TE)-structures is useful. However, it is not clear a priori whether any regular singular
(TE)-structure (H → C×M,∇) over a simply connected manifold M is a good family of
marked regular singular (TE)-structures over X = M . A marking can be imposed as M
is simply connected. Though the condition (3.22) is not clear a priori. Theorem 8.5 will
show this for regular singular rank 2 (TE)-structures. It builds on Theorems 6.3 and 6.7
which show this for regular singular rank 2 (TE)-structures over M = C.
(iii) For not regular singular (TE)-structures, we do not see an easy replacement of condi-
tion (3.22). Is the condition z2∇∂zO(H) ⊂ O(H) useful?
Definition 3.28. Fix a reference pair
(
Href,∞,M ref
)
.
(a) Define the functor M(Href,∞,Mref),reg from the category of complex spaces to the category
of sets by
M(Href,∞,Mref),reg(X) := {(H,ψ) | (H,ψ) is a good family of marked regular
singular (TE)-structures over X},
and, for any morphism f : Y → X of complex spaces and any element (H,ψ) of
M(Href,∞,Mref),reg(X), define M(Href,∞,Mref),reg(f)(H,ψ) := f∗(H,ψ).
(b) Choose r ∈ R and N ∈ N. Define the functor M(Href,∞,Mref),r,N from the category of comp-
lex spaces to the category of sets by
M(Href,∞,Mref),r,N (X) := {(H,ψ) | (H,ψ) is a good family of marked regular
singular(TE)-structures over X which satisfies (3.21)
and (3.22) for the given r and N},
and, for any morphism f : Y → X of complex spaces and any element (H,ψ) of
M(Href,∞,Mref),r,N (X), define M(Href,∞,Mref),r,N (f)(H,ψ) := f∗(H,ψ).
Theorem 3.29. The functors M(Href,∞,Mref),reg and M(Href,∞,Mref),r,N are represented by
complex spaces, which are called M (Href,∞,Mref),reg and M (Href,∞,Mref),r,N . In the case of
M(Href,∞,Mref),r,N , the complex space has even the structure of a projective algebraic variety.
As sets M (Href,∞,Mref),reg = Set(H
ref,∞,Mref),reg.
Proof. The proof for M(Href,∞,Mref),r,N can be copied from the proof of Theorem 7.3 in [15].
Here it is relevant that r and N with (3.21) and (3.22) imply the existence of an r2 ∈ R with
r2 < r and
V r2(x) ⊃ O(H|C×{x})0 for any x ∈ X. (3.23)
In [15], (TERP )-structures are considered. (3.21) and (3.23) are demanded there. (3.22) is not
demanded there explicitly, but it follows from the properties of the pairing there, and this is
used in Lemma 7.2 in [15]. The additional conditions of (TERP )-structures are not essential
for the arguments in the proof of Lemma 7.2 and Theorem 7.3 in [15]. Therefore these proofs
apply also here and give the statements for M(Href,∞,Mref),r,N .
20 C. Hertling
Let us call (r,N) ∈ R × N and
(
r̃, Ñ
)
∈ R × N compatible if n ∈ Z with
(
r̃, Ñ
)
=
(
r + n,
N + n · dimHref,∞) exists. In the case n > 0, M(Href,∞,Mref),r̃,Ñ is a union of M(Href,∞,Mref),r,N
and additional irreducible components. Thus for fixed (r,N) the union⋃
n∈N
M (Href,∞,Mref),r+n,N+n·dimHref,∞
is a complex space with in general countably many irreducible (and compact) components.
And M (Href,∞,Mref),reg is the union of these unions for all possible (r,N) (as Eig(Mmon) is finite,
in each interval of length 1, only finitely many r are relevant). ■
Remarks 3.30.
(i) For each reference pair
(
Href,∞,M ref
)
with dimHref,∞ = 2, the representing complex space
M (Href,∞,Mref),reg for the functorM(Href,∞,Mref),reg is given in Theorem 7.4. There the topo-
logical components are unions
⋃
n∈NM
(Href,∞,Mref),r+n,N+n·dimHref,∞
and have countably
many irreducible components which are either isomorphic to P1 or to the Hirzebruch sur-
face F2 or to the variety F̃2 obtained by blowing down the (−2)-curve in F2. The space
M (Href,∞,Mref),reg is a union of countably many copies of one topological component.
(ii) Corollary 7.3 says that any marked rank 2 regular singular (TE)-structure (H → C ×
M,∇, ψ) with reference pair
(
Href,∞,M ref
)
is a good family of marked regular singular
(TE)-structures. Therefore and because of Theorem 3.29, such a (TE)-structure is induced
by a morphism φ : M → M (Href,∞,Mref),reg. This is crucial for the usefulness of the space
M (Href,∞,Mref),reg. We hope that Corollary 7.3 and this implication are also true for higher
rank regular singular (TE)-structures.
4 Rank 2 (TE)-structures over a point
Here we will classify the rank 2 (TE)-structures over a point.
4.1 Separation into 4 cases
They separate naturally into 4 cases.
Definition 4.1. Let (H → C,∇) be a rank 2 (TE)-structure over a point t0 = 0. Its formal
invariants δ(0), ρ(0), δ(1), ρ(1) from Lemma 3.9 are complex numbers. The eigenvalues of −U are
called u1, u2 ∈ C. They are given by (x − u1)(x − u2) = x2 + 2ρ(0)x + δ(0). We separate four
cases:
(Sem) U has two different eigenvalues −u1 and −u2 ∈ C, i.e., 0 ̸= δ(0) −
(
ρ(0)
)2
.
(Bra) U has only one eigenvalue
(
which is then ρ(0)
)
and one 2 × 2 Jordan block, and δ(1) −
2ρ(0)ρ(1) ̸= 0.
(Reg) U has only one eigenvalue
(
which is then ρ(0)
)
and one 2 × 2 Jordan block, and δ(1) −
2ρ(0)ρ(1) = 0.
(Log) U = ρ(0) · id.
Here (Sem) stands for semisimple, (Bra) for branched, (Reg) for regular singular and (Log) for
logarithmic.
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 21
Remark 4.2. Rank 2 (TE)-structures over a point are richer than the germs of mermorphic
rank 2 vector bundles with a pole of order 2. Though the four types above are closely related
to the formal classification of the latter ones by their slopes (the notion of slopes is developed
for example in [19, Section 5]). In rank 2, three slopes are possible, slope 1, slope 1
2 and slope 0.
Slope 1 corresponds to the type (Sem), slope 1
2 to type (Bra), and slope 0 to the types (Reg)
and (Log).
First we will treat the semisimple case (Sem). Then the cases (Bra), (Reg) and (Log) will be
considered together. Lemma 4.9 will justify the names (Bra) and (Reg). Finally, the three cases
(Bra), (Reg) and (Log) will be treated one after the other. The following lemma gives some first
information. Its proof is straightforward.
Lemma 4.3. Let (H → C,∇) be a rank 2 (TE)-structure over a point. Denote by
(
H̃ →
C, ∇̃
)
the (TE)-structure with trace free pole part with
(
O
(
H̃
)
, ∇̃
)
= (O(H),∇) ⊗ Eρ(0)/z from
Lemma 3.10(b)
(
called
(
H [2] → C,∇[2]
)
there
)
, and denote its invariants from Lemma 3.9
by Ũ , δ̃(0), ρ̃(0), δ̃(1), ρ̃(1). Then
Ũ = U − ρ(0) id,
δ̃(0) = δ(0) −
(
ρ(0)
)2
, ρ̃(0) = 0,
δ̃(1) = δ(1) − 2ρ(0)ρ(1), ρ̃(1) = ρ(1).(
H̃ → C, ∇̃
)
is of the same type (Sem) or (Bra) or (Reg) or (Nil) as (H → C,∇). The following
table characterizes of which type the (TE)-structures (H → C,∇) and
(
H̃ → C, ∇̃
)
are
(Sem) (Bra) (Reg) (Log)
δ̃(0) ̸= 0 δ̃(0) = 0, δ̃(1) ̸= 0 δ̃(0) = δ̃(1) = 0, Ũ ̸= 0 Ũ = 0
Especially, Ũ = 0 implies δ̃(0) = δ̃(1) = 0.
4.2 The case (Sem)
A (TE)-structure over a point with a semisimple endomorphism U with pairwise different eigen-
values is formally isomorphic to a socalled elementary model, and its holomorphic isomorphism
class is determined by its Stokes structure. These two facts are well known. A good reference
is [21, Chapter II, Sections 5 and 6]. The older reference [16] considers only the underlying
meromorphic bundle, so
(
O(H)0 ⊗C{z} C{z}
[
z−1
]
,∇
)
.
In order to formulate the result for rank 2 (TE)-structures more precisely, we need some
notation.
Definition 4.4. Choose numbers u1, u2, α1, α2 ∈ C. Consider the flat bundle H ′ → C∗ with
flat connection ∇ and a basis f = (f1, f2) of global flat multivalued sections f1 and f2 with the
monodromy
f
(
z · e2πi
)
= f(z)
(
e−2πiα1 0
0 e−2πiα2
)
.
The new basis v = (v1, v2) which is defined by
v(z) = f(z)
(
eu1/zzα1 0
0 eu2/zzα2
)
22 C. Hertling
(for some choice of log(z)) is univalued. It defines a (TE)-structure with
z2∇∂zv = v ·B and B =
(
−u1 + zα1 0
0 −u2 + zα2
)
.
This (TE)-structure is called an elementary model. The numbers α1 and α2 are called the regular
singular exponents. The formal invariants δ(0), ρ(0), δ(1), ρ(1) ∈ C of the (TE)-structure and the
tuple (u1, u2, α1, α2) (up to joint exchange of the indices 1 and 2) are equivalent because of
δ(0) −
(
ρ(0)
)2
= −1
4
(u1 − u2)
2, ρ(0) = −u1 + u2
2
, (4.1)
δ(1) − 2ρ(0)ρ(1) =
u1 − u2
2
(α1 − α2), ρ(1) =
α1 + α2
2
. (4.2)
Therefore also the tuple (u1, u2, α1, α2) (up to joint exchange of the indices 1 and 2) is a formal
invariant of the (TE)-structure.
Theorem 4.5.
(a) Any rank 2 (TE)-structure over a point with endomorphism U with two different eigen-
values is formally isomorphic to a unique elementary model in Definition 4.4. Here −u1
and −u2 are the eigenvalues of U .
(b) The (TE)-structure in (a) is up to holomorphic isomorphism determined by the num-
bers u1, u2, α1, α2 and two more numbers s1, s2 ∈ C, the Stokes parameters. It is holo-
morphically isomorphic to the elementary model to which it is formally isomorphic if and
only if s1 = s2 = 0.
(c) Any such tuple (u1, u2, α1, α2, s1, s2) ∈
(
C2 \ {(u1, u1) |u1 ∈ C}
)
× C4 determines such
a (TE)-structure. A second tuple
(
ũ1, ũ2, α̃1, α̃2, s̃1, s̃2
)
̸= (u1, u2, α1, α2, s1, s2) determines
an isomorphic (TE)-structure if and only if
(
ũ1, ũ2, α̃1, α̃2, s̃1, s̃2
)
= (u2, u1, α2, α1, s2, s1).
Part (a) follows for example from [21, Chapter II, Theorem 5.7] together with [21, Chapter II,
Remark 5.8] (Theorem 5.7 considers only the underlying meromorphic bundle; Remark 5.8 takes
care of the holomorphic bundle). For the parts (b) and (c), one needs to deal in detail with the
Stokes structure. We will not do it here, as the semisimple case is not central in this paper.
We refer to [21, Chapter II, Sections 5 and 6] or to [14].
Remarks 4.6.
(i) Malgrange’s unfolding result, Theorem 3.16(c), applies to these (TE)-structures. Such
a (TE)-structure has a unique universal unfolding. The parameters (α1, α2, s1, s2) are
constant, the parameters (u1, u2) are local coordinates on the base space. The base space
is an F -manifold of type A2
1 with Euler field E = u1e1 + u2e2. See Remark 5.3(iii).
(ii) We do not offer normal forms for the (TE)-structures in Theorem 4.5 for three reasons:
(1) As said in (i), the (TE)-structures above unfold uniquely to (TE)-structures over
germs of F -manifolds. In that sense they are easy to deal with. (2) It looks difficult to
write down normal forms. (3) Normal forms should be considered together with the Stokes
parameters, and the corresponding Riemann–Hilbert map from the space of monodromy
data (α1, α2, s1, s2) to a space of parameters for normal forms should be studied. This is
a nontrivial project, which does not fit into the main aims of this paper.
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 23
4.3 Joint considerations on the cases (Bra), (Reg) and (Log)
Notation 4.7. We shall use the following matrices,
C1 := 12, C2 :=
(
0 0
1 0
)
, D :=
(
1 0
0 −1
)
, E :=
(
0 1
0 0
)
,
and the relations between them,
C2
2 = 0, D2 = C1, E2 = 0,
C2D = C2 = −DC2, [C2, D] = 2C2,
C2E =
1
2
(C1 −D), EC2 =
1
2
(C1 +D), [C2, E] = −D,
DE = E = −ED, [D,E] = 2E.
Consider a (TE)-structure (H → C,∇) over a point with U of type (Bra), (Reg) or (Log).
Then U has only one eigenvalue, which is ρ(0) ∈ C. We can and will restrict to C{z}-bases v
of O(H)0 such that the matrix B ∈M2×2(C{z}) with z2∇∂zv = v ·B has the shape
B = b1C1 + b2C2 + zb3D + zb4E with b1, b2, b3, b4 ∈ C{z}. (4.3)
Write as in Remark 3.2 B =
∑
k≥0B
(k)zk with B(k) ∈M2×2(C), and write
bj =
∑
k≥0
b
(k)
j zk with b
(k)
j ∈ C for j ∈ {1, 2}, (4.4)
zbj =
∑
k≥1
b
(k)
j zk with b
(k)
j ∈ C for j ∈ {3, 4}. (4.5)
Then the formal invariants δ(0), ρ(0), δ(1) and ρ(1) of Lemma 3.9 are given by
ρ(0) = b
(0)
1 , ρ(1) = b
(1)
1 ,
δ(0) −
(
ρ(0)
)2
= 0, δ(1) − 2ρ(0)ρ(1) = −b(0)2 b
(1)
4 .
We are in the case (Bra) if b
(0)
2 ̸= 0 and b
(1)
4 ̸= 0, in the case (Reg) if b
(0)
2 ̸= 0 and b
(1)
4 = 0,
and in the case (Log) if b
(0)
2 = 0.
Consider T ∈ GL2(C{z}) and the new basis ṽ = v · T and its matrix B̃ =
∑
k≥0 B̃
(k)zk
with z∇z∂z ṽ = ṽ · B̃. Write
T = τ1C1 + τ2C2 + τ3D + τ4E with τj =
∑
k≥0
τ
(k)
j zk, τ
(k)
j ∈ C.
Then B̃ is determined by (3.11), which is
0 = z2∂zT +B · T − T · B̃
= C1
(
z2∂zτ1 +
(
b1 − b̃1
)
τ1 + z
(
b4 − b̃4
)τ2
2
+ z
(
b3 − b̃3
)
τ3 +
(
b2 − b̃2
)τ4
2
)
+ C2
(
z2∂zτ2 +
(
b2 − b̃2
)
τ1 +
(
b1 − b̃1
)
τ2 + z
(
−b3 − b̃3
)
τ2 +
(
b2 + b̃2
)
τ3
)
+D
(
z2∂zτ3 + z
(
b3 − b̃3
)
τ1 + z
(
b4 + b̃4
)τ2
2
+
(
b1 − b̃1
)
τ3 +
(
−b2 − b̃2
)τ4
2
)
+ E
(
z2∂zτ4 + z
(
b4 − b̃4
)
τ1 + z
(
−b4 − b̃4
)
τ3 +
(
b1 − b̃1
)
τ4 + z
(
b3 + b̃3
)
τ4
)
. (4.6)
We will use this quite often in order to construct or compare normal forms. The following
immediate corollary of the proof of Lemma 3.11 provides a reduction of b1.
24 C. Hertling
Corollary 4.8. The base change matrix T = eg · C1 with g as in (3.15) leads to b̃j with
b̃1 = b
(0)
1 + zb
(1)
1 = ρ(0) + zρ(1), b̃2 = b2, b̃3 = b3, b̃4 = b4,
From now on we will work in this section only with bases v with b1 = ρ(0) + zρ(1). This is
justified by Corollary 4.8.
Furthermore, we will consider from now on in this section mainly (TE)-structures with trace
free pole part
(
Definition 3.8, ρ(0) = 1
2 trU = 0
)
. See Lemmata 3.10 and 3.11 for the relation to
the general case.
The next lemma separates the cases (Bra) and (Reg).
Lemma 4.9. Consider a (TE)-structure over a point with U of type (Bra) or type (Reg) and
with trace free pole part (so U is nilpotent but not 0).
The (TE)-structure is regular singular if and only if it is of type (Reg). If it is of type
(Bra), then the pullback of O(H)0 ⊗C{z} C{z}
[
z−1
]
by the map C → C, x 7→ x4 = z, is the
space of germs at 0 of sections of a meromorphic bundle on C with a meromorphic connection
with an order 3 pole at 0 with semisimple pole part with eigenvalues κ1 and κ2 = −κ1 with
−1
4κ
2
1 = δ(1) ∈ C∗. Thus κ21 is a formal invariant of the (TE)-structure of type (Bra).
Proof. Consider a C{z}-basis v of O(H)0 such that its matrix B is as in (4.3) and such that
b1 = zρ(1). This is possible by Corollary 4.8 and the assumption ρ(0) = 0. As U is nilpotent,
but not 0, b
(0)
2 ̸= 0. Now δ(1) = −b(0)2 b
(1)
4 , so δ(1) ̸= 0 ⇐⇒ b
(1)
4 ̸= 0.
Consider the case b
(1)
4 ̸= 0, and consider the pullback of the (TE)-structure by the map
C → C, x 7→ x4 = z. Then dz
z = 4dx
x and z∂z =
1
4x∂x and
∇x∂xv = v · 4
∑
k≥0
B(k)x4k−4,
∇x∂x
(
v · xD
)
=
(
v · xD
)
4
×
(
x−2
∑
k≥0
(
b
(k)
2 C2 + b
(k+1)
4 E
)
x4k+ ρ(1)C1 +
(
1
4
+
∑
k≥0
b
(k+1)
3 x4k
)
D
)
. (4.7)
One sees a pole of order 3 with matrix 4
(
b
(0)
2 C2+ b
(1)
4 E
)
of the pole part. It is tracefree and has
the eigenvalues κ1 and κ2 = −κ1 with κ21 = 4b
(0)
2 b
(1)
4 ∈ C∗. This shows the claims in the case
b
(1)
4 ̸= 0.
Consider the case b
(1)
4 = 0, and consider the pullback of the (TE)-structure by the map
C → C, x 7→ x2 = z. Then dz
z = 2dx
x and z∂z =
1
2x∂x and
∇x∂xv = v · 2
∑
k≥0
B(k)x2k−2,
∇x∂x
(
v · xD
)
=
(
v · xD
)
2
×
(
ρ(1)C1 +
1
2
D +
∑
k≥0
(
b
(k)
2 C2 + b
(k+2)
4 E + b
(k+1)
3 D
)
x2k
)
. (4.8)
One sees a logarithmic pole. Therefore also the sections v1 and v2 have moderate growth, and the
(TE)-structure is regular singular. ■
Remark 4.10. The two transformations in (4.7) (for the case (Bra)) and (4.8) (for the case
(Reg)) are special cases of a systematic development of such ramified gauge transformations
in [2] (a short description is given in [24, p. 17]). The basic idea goes back to the shearing
transformations of Fabry (see [8] and [24, p. 4]).
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 25
4.4 The case (Bra)
The following theorem gives complete control on the (TE)-structures over a point of the type
(Bra). Here Eig(Mmon) ⊂ C is the set of eigenvalues of the monodromy of such a (TE)-structure
(it has 1 or 2 elements).
Theorem 4.11.
(a) Consider a (TE)-structure over a point of the type (Bra). The formal invariants ρ(0), ρ(1)
and δ(1) ∈ C from Lemma 3.9 and the set Eig(Mmon) are invariants of its isomorphism
class. Together they form a complete set of invariants. That means, the isomorphism class
of the (TE)-structure is determined by these invariants.
(b) Any such (TE)-structure has a C{z}-basis v of O(H)0 such that its matrix is in Birkhoff
normal form, and more precisely, the matrix B has the shape
B =
(
ρ(0) + zρ(1)
)
C1 + b
(0)
2 C2 + zb
(1)
3 D + zb
(1)
4 E, (4.9)
where b
(0)
2 , b
(1)
4 ∈ C∗ and b
(1)
3 ∈ C satisfy −b(0)2 b
(1)
4 = δ(1) − 2ρ(0)ρ(1) and Eig(Mmon) ={
e−2πi(ρ(1)±b
(1)
3 )
}
.
Remarks 4.12.
(i) Because of part (a), two Birkhoff normal forms as in (4.9) with data
(
ρ(0), ρ(1), b
(0)
2 ,
b
(1)
3 , b
(1)
4
)
and
(
ρ̃(0), ρ̃(1), b̃
(0)
2 , b̃
(1)
3 , b̃
(1)
4
)
give isomorphic (TE)-structures if and only if ρ̃(0) =
ρ(0), ρ̃(1) = ρ(1), b̃
(0)
2 b̃
(1)
4 = b
(0)
2 b
(1)
4 and b̃
(1)
3 ∈
{
±b(1)3 +k | k ∈ Z
}
. However, the pure (TLE)-
structures which they define, are isomorphic only if additionally b̃
(1)
3 ∈
{
±b(1)3
}
.
(ii) We could restrict to Birkhoff normal forms with b
(0)
2 = 1 or with b
(1)
4 = 1. Though in view
of the (TE)-structures in the 4th case in Theorem 6.3 we prefer not to do that.
Proof of Theorem 4.11. The proof has 3 steps.
Step 1: Birkhoff normal forms exist. In order to show this, it is sufficient to prove the hypothesis
in Theorem 3.20(b). The hypothesis says that the germ of the meromorphic bundle underlying
a (TE)-structure of type (Bra) is irreducible. A proof by calculation is given in the proof
of Lemma 28 in [6]. Though this is also well known as the rank is two and the slope is 1
2 (see
Remark 4.2).
Step 2: Analysis of the Birkhoff normal forms. The matrix B of a Birkhoff normal form can be
chosen with b1 = ρ(0) + zρ(1) because of Corollary 4.8. Then it has the shape
B =
(
ρ(0) + zρ(1)
)
C1 +
(
b
(0)
2 + zb
(1)
2
)
C2 + zb
(1)
3 D + zb
(1)
4 E
with b
(0)
2 ̸= 0 and b
(1)
4 ̸= 0.
Consider the new basis ṽ = v · T and its matrix B̃, where
T = C1 + τ
(0)
2 C2 for some τ
(0)
2 ∈ C. (4.10)
Equation (4.6) gives
0 =
(
b1 − b̃1
)
+ z
(
b
(1)
4 − b̃4
)τ (0)2
2
, 0 =
(
b2 − b̃2
)
+
(
b1 − b̃1
)
τ
(0)
2 + z
(
−b(1)3 − b̃3
)
τ
(0)
2 ,
0 =
(
b
(1)
3 − b̃3
)
+
(
b
(1)
4 + b̃4
)τ (0)2
2
, 0 =
(
b
(1)
4 − b̃4
)
,
26 C. Hertling
so
b̃4 = b̃
(1)
4 = b
(1)
4 , b̃1 = b1, b̃3 = b̃
(1)
3 = b
(1)
3 + b
(1)
4 τ
(0)
2 ,
b̃
(0)
2 = b
(0)
2 , b̃
(1)
2 = b
(1)
2 − 2b
(1)
3 τ
(0)
2 − b
(1)
4
(
τ
(0)
2
)2
. (4.11)
τ
(0)
2 can be chosen such that b̃
(1)
2 = 0. Then the Birkhoff normal form B̃ has the shape in (4.9).
Suppose now that B has this shape, so b2 = b
(0)
2 . The choice τ
(0)
2 := −2b
(1)
3 /b
(1)
4 in (4.10)
leads to
b̃1 = b1, b̃2 = b2, b̃4 = b4 and b̃3 = −b3. (4.12)
Consider the new basis ṽ = v · T and its matrix B̃, where
T = C1 + τ
(0)
3 D for some τ
(0)
3 ∈ C \ {±1}.
Equation (4.6) gives
0 =
(
b1 − b̃1
)
+ z
(
b
(1)
3 − b̃3
)
τ
(0)
3 ,
0 =
(
b
(0)
2 − b̃2
)
+
(
b
(0)
2 + b̃2
)
τ
(0)
3 ,
0 = z
(
b
(1)
3 − b̃3
)
+
(
b1 − b̃1
)
τ
(0)
3 ,
0 =
(
b
(1)
4 − b̃4
)
+
(
−b(1)4 − b̃4
)
τ
(0)
3 ,
so
b̃1 = b1, b̃3 = b
(1)
3 , b̃2 = b
(0)
2
1 + τ
(0)
3
1− τ
(0)
3
, b̃4 = b
(1)
4
1− τ
(0)
3
1 + τ
(0)
3
. (4.13)
So, in a Birkhoff normal form in (4.9), one can change b
(0)
2 and b
(1)
4 arbitrarily with constant
product b
(0)
2 b
(1)
4 and without changing b1 = ρ(0) + zρ(1) and b
(1)
3 .
Consider the new basis ṽ = v · T and its matrix B̃, where
T =
(
1 + zτ
(1)
1
)
C1 + τ
(0)
2 C2 + zτ
(1)
3 D + zτ
(1)
4 E, for some τ
(1)
1 , τ
(0)
2 , τ
(1)
3 , τ
(1)
4 ∈ C.
We are searching for coefficients τ
(1)
1 , τ
(0)
2 , τ
(1)
3 , τ
(1)
4 ∈ C such that
b̃1 = b1, b̃2 = b2, b̃4 = b4, b̃3 = b3 + ε with ε = ±1. (4.14)
Under these constraints, (4.6) gives
0 = τ
(1)
1 − ετ
(1)
3 ,
0 =
(
−2b
(1)
3 − ε
)
τ
(0)
2 + 2b
(0)
2 τ
(1)
3 ,
0 = zτ
(1)
3 − ε
(
1 + zτ
(1)
1
)
+ b
(1)
4 τ
(0)
2 − b
(0)
2 τ
(1)
4 ,
0 = τ
(1)
4 − 2b
(1)
4 τ
(1)
3 +
(
2b
(1)
3 + ε
)
τ
(1)
4 .
With τ
(1)
1 = ετ
(1)
3 , these equations boil down to the inhomogeneous linear system of equations0
ε
0
=
−2b
(1)
3 − ε 2b
(0)
2 0
b
(1)
4 0 −b(0)2
0 −2b
(1)
4 2b
(1)
3 + ε+ 1
τ
(0)
2
τ
(1)
3
τ
(1)
4
. (4.15)
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 27
The determinant of the 3× 3 matrix is −2b
(0)
2 b
(1)
4 ̸= 0. Therefore the system (4.15) has a unique
solution
(
τ
(0)
2 , τ
(1)
3 , τ
(1)
4
)t
. Thus a new basis ṽ = v · T with (4.14) exists.
Iterating this construction, one finds that one can change the matrix B in (4.9) by a holo-
morphic base change to a matrix B̃ with
b̃1 = b1, b̃2 = b2, b̃4 = b4, b̃3 = b3 + k, (4.16)
for any k ∈ Z.
Putting together (4.11), (4.12), (4.13) and (4.16), one sees that two Birkhoff normal forms as
in (4.9) with data
(
ρ(0), ρ(1), b
(0)
2 , b
(1)
3 , b
(1)
4
)
and
(
ρ̃(0), ρ̃(1), b̃
(0)
2 , b̃
(1)
3 , b̃
(1)
4
)
give isomorphic (TE)-
structures if ρ̃(0) = ρ(0), ρ̃(1) = ρ(1), b̃
(0)
2 b̃
(1)
4 = b
(0)
2 b
(1)
4 and b̃
(1)
3 ∈
{
± b
(1)
3 + k | k ∈ Z
}
. This shows
if in Remark 4.12(i).
Step 3: Discussion of the invariants. By Lemma 3.9, ρ(0), ρ(1) and δ(1) are even formal invariants
of the (TE)-structure. The set Eig(Mmon) is obviously an invariant of the isomorphism class of
the (TE)-structure.
The Birkhoff normal form in (4.9) gives a pure (TLE)-structure with a logarithmic pole at ∞.
From its pole part at ∞ and Theorem 3.23(c) one reads off
Eig(Mmon) =
{
e−2πi(ρ(1)±b
(1)
3 )
}
.
As ρ(1) is an invariant of the (TE)-structure, also the set
{
± b
(1)
3 + k | k ∈ Z
}
is an invariant of
the (TE)-structure.
Together with Step 2, this shows only if in Remark 4.12(i) and all statements in Theo-
rem 4.11. ■
Corollary 4.13. The monodromy of a (TE)-structure over a point of the type (Bra) has a 2×2
Jordan block if its eigenvalues coincide
(
equivalently, if b
(1)
3 ∈ 1
2Z for some (or any) Birkhoff
normal form in Theorem 4.11(b)
)
.
Proof. Consider a (TE)-structure over a point of the type (Bra) such that the eigenvalues of
its monodromy coincide. Then for any Birkhoff normal form in Theorem 4.11(b) b
(1)
3 ∈ 1
2Z, and
one can choose a Birkhoff normal form with b
(1)
3 ∈
{
0,−1
2
}
. The induced pure (TLE)-structure
has at ∞ a logarithmic pole, and its residue endomorphism [∇z̃∂z̃ ], where z̃ = z−1, is given by
the matrix −
(
ρ(1)C1 + b
(1)
3 D + b
(1)
4 E
)
.
In the case b
(1)
3 = 0, the nonresonance condition in Theorem 3.23(c) is satisfied, so Theo-
rem 3.23(c) can be applied. Because of b
(1)
4 ̸= 0, the monodromy has a 2× 2 Jordan block.
In the case b
(1)
3 = −1
2 , the meromorphic base change
ṽ := v ·
(
z 0
0 1
)
gives the new connection matrix
B̃ =
(
ρ(0) + z
(
ρ(1) +
1
2
))
C1 + zb
(0)
2 C2 + b
(1)
4 E.
Again, the pole at ∞ is logarithmic. Now the nonresonance condition in Theorem 3.23(c) is
satisfied. Because of b
(0)
2 ̸= 0, the monodromy has a 2× 2 Jordan block. ■
For (TE)-structures of the type (Bra), formal isomorphism is coarser than holomorphic iso-
morphism.
28 C. Hertling
Lemma 4.14. Consider a (TE)-structure over a point of the type (Bra). By Lemma 3.9, the
numbers ρ(0), ρ(1) and δ(1) are formal invariants of the (TE)-structure.
(a) The set Eig(Mmon) and the equivalent set
{
± b
(1)
3 + k | k ∈ Z
}
are holomorphic invariants,
but not formal invariants.
(b) The (TE)-structure with Birkhoff normal form in (4.9) is formally isomorphic to the (TE)-
structure with Birkhoff normal form in (4.9) with the same values ρ(0), ρ(1), b
(0)
2 and b
(1)
4 ,
but with an arbitrary b̃
(1)
3 .
Proof. Part (a) follows from part (b). For the proof of part (b), we have to find T ∈ GL2(C[[z]])
such that T , B in (4.9) and
B̃ =
(
ρ(0) + zρ(1)
)
C1 + b
(0)
2 C2 + zb̃
(1)
3 D + zb
(1)
4 E
with b̃
(1)
3 ∈ C arbitrary satisfy (4.6). Here (4.6) says
0 = z∂zτ1 +
(
b
(1)
3 − b̃
(1)
3
)
τ3,
0 = z2∂zτ2 + z
(
−b(1)3 − b̃
(1)
3
)
τ2 + 2b
(0)
2 τ3,
0 = z2∂zτ3 + z
(
b
(1)
3 − b̃
(1)
3
)
τ1 + zb
(1)
4 τ2 − b
(0)
2 τ4,
0 = z∂zτ4 − 2b
(1)
4 τ3 +
(
b
(1)
3 + b̃
(1)
3
)
τ4.
This is equivalent to
0 = τ
(0)
3 = τ
(0)
4 ,
0 = kτ
(k)
1 +
(
b
(1)
3 − b̃
(1)
3
)
τ
(k)
3 for k ≥ 1,
0 =
(
k − 1− b
(1)
3 − b̃
(1)
3
)
τ
(k−1)
2 + 2b
(0)
2 τ
(k)
3 for k ≥ 1,
0 = (k − 1)τ
(k−1)
3 +
(
b
(1)
3 − b̃
(1)
3
)
τ
(k−1)
1 + b
(1)
4 τ
(k−1)
2 − b
(0)
2 τ
(k)
4 for k ≥ 1,
0 = −2b
(1)
4 τ
(k)
3 +
(
k + b
(1)
3 + b̃
(1)
3
)
τ
(k)
4 for k ≥ 1.
This is equivalent to
τ
(0)
3 = τ
(0)
4 = 0,
τ
(k)
1 =
−1
k
(
b
(1)
3 − b̃
(1)
3
)
τ
(k)
3 for k ≥ 1,
2b
(0)
2 τ
(k)
3 =
(
b
(1)
3 + b̃
(1)
3 + 1− k
)
τ
(k−1)
2 for k ≥ 1,
b
(0)
2 τ
(1)
4 = b
(1)
4 τ
(0)
2 +
(
b
(1)
3 − b̃
(1)
3
)
τ
(0)
1 ,
b
(0)
2 τ
(k)
4 = b
(1)
4 τ
(k−1)
2 +
(
k − 1 +
−1
k − 1
(
b
(1)
3 − b̃
(1)
3
)2)(
2b
(0)
2
)−1(
b
(1)
3 + b̃
(1)
3 + 2− k
)
τ
(k−2)
2
for k ≥ 2,
0 = b
(1)
4 τ
(0)
2 +
(
1 + b
(1)
3 + b̃
(1)
3
)(
b
(1)
3 − b̃
(1)
3
)
τ
(0)
1 ,
0 = b
(1)
4 (2k + 1)τ
(k)
2 +
(
k + 1 + b
(1)
3 + b̃
(1)
3
)(
k +
−1
k
(
b
(1)
3 − b̃
(1)
3
)2)(
2b
(0)
2
)−1
×
(
b
(1)
3 + b̃
(1)
3 + 1− k
)
τ
(k−1)
2 for k ≥ 1. (4.17)
One can choose τ
(0)
1 ∈ C∗ freely. Then the equations (4.17) have unique solutions τ1 − τ
(0)
1 , τ2,
τ3, τ4 ∈ C[[z]]. Therefore T ∈ GL2(C[[z]]) exists such that T , B as in (4.9) and B̃ as above
satisfy (4.6). This shows part (b). ■
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 29
Remarks 4.15.
(i) Because of Lemma 4.14, the set Eig(Mmon) takes here the role of the Stokes structure:
It distinguishes the holomorphic isomorphism classes within one formal isomorphism class.
(ii) The following (TE)-structures have trivial Stokes structure. The proof of Lemma 4.9 leads
to these (TE) structures. They are the (TE)-structures with Eig(Mmon) = {λ1, λ2} with
λ2 = −λ1, respectively with b
(1)
3 ∈
(
± 1
4 + Z
)
for any Birkhoff normal form in (4.9).
Choose a number α(1) ∈ C. Consider the rank 2 bundle H ′ → C∗ with flat connection ∇
and flat multivalued basis f = (f1, f2) with monodromy given by
f(ze2πi) = f(z) · ie−2πiα(1) · (C2 + E).
The eigenvalues are ±ie−2πiα(1)
. Choose numbers t1 ∈ C and t2 ∈ C∗. The following basis
of H ′ is univalued.
v := f · et1/zzα(1)
(
z−1/4et2z
−1/2
z1/4et2z
−1/2
z−1/4e−t2z−1/2 −z1/4e−t2z−1/2
)
. (4.18)
The matrix B with z2∇∂zv = v ·B is
B =
(
−t1 + zα(1)
)
C1 −
t2
2
C2 − z
1
4
D − z
t2
2
E. (4.19)
So here ρ(1) = α(1), ρ(0) = −t1, δ(1) − 2ρ(0)ρ(1) = −1
4 t
2
2.
(iii) Part (ii) generalizes to a (TE)-structure over M = C2 with coordinates t = (t1, t2).
Consider the rank 2 bundle H ′ → C∗ ×M with flat connection and flat multivalued basis
f = (f1, f2) with monodromy given by
f
(
ze2πi, t
)
= f(z, t) · ie−2πiα(1) · (C2 + E).
The basis v in (4.18) is univalued. The matrices A1, A2 and B in its connection 1-form Ω
as in (3.2)–(3.4) are given by (4.19) and
A1 = C1, A2 = C2 + zE.
The restriction to a point t ∈ C×C∗ is a (TE)-structure of type (Bra) with trivial Stokes
structure. The restriction to a point t ∈ C× {0} is a (TE)-structure of type (Log).
4.5 The case (Reg) with trU = 0
The (TE)-structures over a point of the type (Reg) with trU = 0 are the regular singular (TE)-
structures over a point which are not logarithmic. They can be easily classified using elementary
sections. Theorem 4.17 splits them into three cases (one in part (a), two in part (b): α1 = α2
and α1 − α2 ∈ N).
Notation 4.16. Start with a (TE)-structure (H → C,∇) of rank 2 over a point. Recall
the notions from Definition 3.21: H ′ := H|C∗ , Mmon, Mmon
s , Mmon
u , Nmon, Eig(Mmon), H∞,
H∞
λ , Cα for α ∈ C with e−2πiα ∈ Eig(Mmon), s(A,α) ∈ Cα for A ∈ H∞
e−2πiα , es(σ, α) ∈ Cα for σ
a holomorphic section on H|U1\{0} for U1 ⊂ C a neighborhood of 0. Now the eigenvalues ofMmon
30 C. Hertling
are called λ1 and λ2 (λ1 = λ2 is allowed). The sheaf V>−∞ simplifies here to a C{z}
[
z−1
]
-vector
space of dimension 2,
V >−∞ :=
C{z}
[
z−1
]
· Cα1 ⊕ C{z}
[
z−1
]
· Cα2 if λ1 ̸= λ2,
C{z}
[
z−1
]
· Cα1 if λ1 = λ2,
where α1, α2 ∈ C with e−2πiαj = λj . V
>−∞ is the space of sections of moderate growth.
Theorem 4.17. Consider a regular singular, but not logarithmic, rank 2 (TE)-structure (H →
C,∇) over a point. Associate to it the data in the Notation 4.16.
(a) The case Nmon = 0: There exist unique numbers α1, α2 with e−2πiαj = λj and α1 ̸= α2 and
the following properties: There exist elementary sections s1 ∈ Cα1 \{0} and s2 ∈ Cα2 \{0}
and a number t2 ∈ C∗ such that
O(H)0 = C{z}(s1 + t2s2)⊕ C{z}(zs2) (4.20)
= C{z}
(
s2 + t−1
2 s1
)
⊕ C{z}(zs1). (4.21)
The isomorphism class of the (TE)-structure is uniquely determined by the information
Nmon = 0 and the set {α1, α2}. The numbers α1 and α2 are called leading exponents.
(b) The case Nmon ̸= 0 (thus λ1 = λ2): There exist unique numbers α1, α2 with e−2πiαj = λ1
and α1 − α2 ∈ N0 and the following properties: Choose any elementary section s1 ∈
Cα1 \ ker(z∇∂z − α1 : C
α1 → Cα1). The elementary section s2 ∈ Cα2 with
(z∇∂z − α1)(s1) = zα1−α2s2. (4.22)
is a generator of ker(z∇∂z − α2 : C
α2 → Cα2). Then
O(H)0 = C{z}(s1 + t2s2)⊕ C{z}(zs2) (4.23)
for some t2 ∈ C. If α1 > α2 then t2 is in C∗ and is independent of the choice of s1.
If α1 = α2, then one can replace s1 by s
[new]
1 := s1 + t2s2, and then t
[new]
2 = 0. The iso-
morphism class of the (TE)-structure is uniquely determined by the information Nmon ̸= 0
and the pair (α1, α2) if α1 = α2 and the triple (α1, α2, t2) if α1 > α2. The numbers α1
and α2 are called leading exponents.
Proof. First, (a) and (b) are considered together. Let β1, β2 ∈ C be the unique numbers with
e−2πiβj = λj and −1 < Re(βj) ≤ 0. Choose elementary sections s̃1 ∈ Cβ1 and s̃2 ∈ Cβ2
which form a global basis of H ′. In the case Nmon ̸= 0 (then β1 = β2) choose them such that
s̃1 /∈ ker
(
z∇∂z − β1 : C
β1 → Cβ1
)
and s̃2 ∈ ker
(
z∇∂z − β2 : C
β2 → Cβ2
)
.
Let σ
[1]
1 , σ
[1]
2 ∈ O(H)0 be a C{z}-basis of O(H)0. Write
(
σ
[1]
1 , σ
[1]
2
)
=
(
s̃1, s̃2
)(b11 b12
b21 b22
)
with bij ∈ C{z}
[
z−1
]
.
Recall that the degree degz g of a Laurent series g =
∑
j∈Z g
(j)zj ∈ C{z}
[
z−1
]
is the minimal j
with g(j) ̸= 0 if g ̸= 0, and degz 0 := +∞.
In the case Nmon = 0 and λ1 = λ2 (then β1 = β2), we suppose min(degz b11, degz b12) ≤
min(degz b21,degz b22). If it does not hold a priori, we can exchange s̃1 and s̃2.
In any case, we suppose degz b11 ≤ degz b12. If it does not hold a priori, we can exchange σ
[1]
1
and σ
[1]
2 .
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 31
Again in the case Nmon = 0 and λ1 = λ2, we suppose degz b11 < degz b21. If it does not hold
a priori, we can replace s̃2 by a certain linear combination of s̃2 and s̃1.
Now b̃11 := z− degz b11b11 ∈ C{z}∗ is a unit. Consider α1 := β1 +degz b11 and s1 := zdegz b11 s̃1
∈ Cα1 and the new basis
(
σ
[2]
1 , σ
[2]
2
)
of O(H)0 with
(
σ
[2]
1 , σ
[2]
2
)
:=
(
σ
[1]
1 , σ
[1]
2
)(b̃−1
11 −b−1
11 b12
0 1
)
=
(
s1, s̃2
)( 1 0
b̃−1
11 b21 b22 − b−1
11 b12b21
)
.
Consider m := degz
(
b22 − b−1
11 b12b21
)
∈ Z (+∞ is impossible) and α2 := β2 + (m − 1) and
s2 := zm−1s̃2 ∈ Cα2 . Write z−m+1b̃−1
11 b21 = c1 + c2 with c1 ∈ C
[
z−1
]
and c2 ∈ zC{z}. We can
replace σ
[2]
2 by σ
[3]
2 := zs2 and σ
[2]
1 = s1 + (c1 + c2)s2 by σ
[3]
1 = s1 + c1s2.
(a) Consider the case Nmon = 0. If λ1 = λ2 then degz b21 ≥ degz b11 + 1 and thus
(c1 + c2)s2 = b̃−1
11 b21s̃2 ∈ C{z} · zdegz b21 · Cβ2
⊂ C{z} · zdegz b11+1 · Cβ2 = C{z} · Cα1+1. (4.24)
In any case (whether λ1 = λ2 or λ1 ̸= λ2), we must have c1 ̸= 0. Else the (TE)-structure
is logarithmic. As the pole has precisely order 2, c1 is a constant ̸= 0 (if λ1 = λ2, here we
need (4.24)), which is now called t2. This implies m− 1 = degz b21. In the case Nmon = 0 and
λ1 = λ2 we have β1 = β2 and
α2 − α1 = m− 1− degz b11 = degz b21 − degz b11 > 0,
so especially α2 ̸= α1.
(b) Consider the case Nmon ̸= 0. Then s2 is generator of ker(z∇∂z −α2 : C
α2 → Cα2), and we
can rescale it such that (4.22) holds. First consider the case c1 = 0. As the pole has precisely
order 2, we must have α2 = α1. Then (4.23) holds with t2 = 0. Now consider the case c1 ̸= 0.
Then σ[3] =
(
σ
[3]
1 , σ
[3]
2
)
satisfies
z∇∂zσ
[3] = σ[3]
(
α1 0
z−1(z∂z − α1 + α2)(c1) + zα1−α2−1 α2 + 1
)
.
First case, α1−α2 ∈ Z<0: The coefficient of zα1−α2−1 in z−1(z∂z −α1+α2)(c1)+ z
α1−α2−1 is 1.
Therefore the pole order is > 2, a contradiction.
Second case, α1 ≥ α2: As the pole has precisely order 2, c1 is a constant ̸= 0, which is now
called t2. Then (4.23) holds, and t2 ∈ C∗. In the case α1 − α2 ∈ N, t2 is obviously independent
of the choice of s1. ■
Corollary 4.18 is an immediate consequence of Theorem 4.17.
Corollary 4.18. The set of regular singular, but not logarithmic, rank 2 (TE)-structures over
a point is in bijection with the set
{(0, {α1, α2}) |α1, α2 ∈ C, α1 ̸= α2} ∪ {(1, α1, α2) |α1 = α2 ∈ C}
∪ {(1, α1, α2, t2) |α1, α2 ∈ C, α1 − α2 ∈ N, t2 ∈ C∗}.
The first set parametrizes the cases with Nmon = 0, the second and third set parametrize the
cases with Nmon ̸= 0. Theorem 4.17 describes the corresponding (TE)-structures.
Remark 4.19. The connection matrices for the special bases in Theorem 4.17 can be written
down easily.
32 C. Hertling
The basis in (4.20):
∇z∂z(s1 + t2s2, zs2) = (s1 + t2s2, zs2)
(
α1 0
z−1(α2 − α1)t2 α2 + 1
)
. (4.25)
The basis in (4.21) with t̃2 := t−1
2 :
∇z∂z
(
s2 + t̃2s1, zs1
)
=
(
s2 + t̃2s1, zs1
)( α2 0
z−1(α1 − α2)t̃2 α1 + 1
)
. (4.26)
The basis in (4.23) with (4.22):
∇z∂z(s1 + t2s2, zs2) = (s1 + t2s2, zs2)
(
α1 0
z−1(α2 − α1)t2 + zα1−α2−1 α2 + 1
)
. (4.27)
Finally, in the case Nmon ̸= 0 and t2 ∈ C∗, we consider with t̃2 := t−1
2 also the basis(
s2 + t̃2s1, zs1
)
. Again (4.22) is assumed:
∇z∂z
(
s2 + t̃2s1, zs1
)
=
(
s2 + t̃2s1, zs1
)
×
(
α2 + zα1−α2 t̃2 zα1−α2+1
z−1(α1 − α2)t̃2 − zα1−α2−1t̃22 α1 + 1− zα1−α2 t̃2
)
. (4.28)
4.6 The case (Log) with trU = 0
The (TE)-structures over a point of the type (Log) with trU = 0 are the logarithmic (TE)-
structures over a point. Just as the regular singular (TE)-structures, they can easily be classified
using elementary sections. Theorem 4.20 splits them into two cases. We use again the Nota-
tion 4.16.
Theorem 4.20. Consider a logarithmic rank 2 (TE)-structure (H → C,∇) over a point. Asso-
ciate to it the data in the Notation 4.16.
(a) The case Nmon = 0: There exist unique numbers α1, α2 with e−2πiαj = λj and the following
property: There exist elementary sections s1 ∈ Cα1 \ {0} and s2 ∈ Cα2 \ {0} such that
O(H)0 = C{z} s1 ⊕ C{z} s2. (4.29)
The isomorphism class of the (TE)-structure is uniquely determined by the information
Nmon = 0 and the set {α1, α2}. The numbers α1 and α2 are called leading exponents.
(b) The case Nmon ̸= 0 (thus λ1 = λ2): There exist unique numbers α1, α2 with e−2πiαj = λ1
and α1 − α2 ∈ N0 and the following properties: Choose any elementary section s1 ∈
Cα1 − ker(∇z∂z − α1 : C
α1 → Cα1). The elementary section s2 ∈ Cα2 with
(z∇∂z − α1)(s1) = zα1−α2s2. (4.30)
is a generator of ker(z∇∂z − α2 : C
α2 → Cα2). Then
O(H)0 = C{z} s1 ⊕ C{z} s2. (4.31)
The isomorphism class of the (TE)-structure is uniquely determined by the information
Nmon ̸= 0 and the set {α1, α2}. The numbers α1 and α2 are called leading exponents.
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 33
Proof. First, (a) and (b) are considered together. By Theorem 3.23(a), O(H)0 is generated by
two elementary sections s1 ∈ Cα1 and s2 ∈ Cα2 for some numbers α1 and α2. The numbers α1
and α2 are the eigenvalues of the residue endomorphism. So, they are unique. This finishes
already the proof of part (a).
(b) Consider the case Nmon ̸= 0. We can renumber s1 and s2 if necessary, so that afterwards
α1 − α2 ∈ N0. If α1 = α2, then O(H)0 = C{z}Cα1 , and s1 and s2 can be changed so that
s1 ∈ Cα1 \ ker(∇z∂z − α1) and s2 ∈ ker(∇z∂z − α1 : C
α1 → Cα1) \ {0} satisfy (4.30). Then
nothing more has to be shown. Consider the case α1−α2 ∈ N. If s2 ∈ Cα2 \ker(∇z∂z −α2), then
(∇z∂z −α2)(s2) is not in O(H)0, and thus the pole is not logarithmic, a contradiction. Therefore
s2 ∈ ker(∇z∂z − α2 : C
α2 → Cα2). Then necessarily s1 ∈ Cα1 \ ker(∇z∂z − α1 : C
α1 → Cα1).
We can rescale s2 so that (4.30) holds. Nothing more has to be shown. ■
Corollary 4.21 is an immediate consequence of Theorem 4.20.
Corollary 4.21. The set of logarithmic rank 2 (TE)-structures over a point is in bijection with
the set
{(0, {α1, α2}) |α1, α2 ∈ C, } ∪ {(1, α1, α2) |α1, α2 ∈ C, α1 − α2 ∈ N0}.
The first set parametrizes the cases with Nmon = 0, the second set parametrizes the cases with
Nmon ̸= 0. Theorem 4.20 describes the corresponding (TE)-structures.
Remark 4.22. The connection matrices for the special bases in Theorem 4.20 can be written
down easily.
The basis in (4.29):
∇z∂z(s1, s2) = (s1, s2)
(
α1 0
0 α2
)
.
The basis in (4.31):
∇z∂z(s1, s2) = (s1, s2)
(
α1 0
zα1−α2 α2
)
.
The basis (s1, s2) gives a Birkhoff normal form in the cases Nmon = 0 and in the cases
(Nmon ̸= 0 and α1 = α2). In the cases (Nmon ̸= 0 and α1 − α2 ∈ N), a Birkhoff normal form
does not exist.
5 Rank 2 (TE)-structures over germs of regular F -manifolds
This section discusses unfoldings of (TE)-structures over a point t0 of type (Sem) or (Bra)
or (Reg). Here Malgrange’s unfolding result Theorem 3.16(c) applies. It provides a universal
unfolding for the (TE)-structure over t0. Any unfolding is induced by the universal unfolding.
The universal unfoldings turn out to be precisely the (TE)-structures with primitive Higgs fields
over germs of regular F -manifolds.
Sections 6 and 8 discuss unfoldings of (TE)-structures over a point of type (Log). Section 8
treats arbitrary such unfoldings. Section 6 prepares this. It treats 1-parameter unfoldings with
trace free pole parts of logarithmic (TE)-structures over a point.
If one starts with a (TE)-structure with primitive Higgs field over a germ
(
M, t0
)
of a regular
F -manifold, then the endomorphism U|t0 : Kt0 → Kt0 is regular.
Vice versa, if one starts with a (TE)-structure over a point t0 with a regular endomorphism
U : Kt0 → Kt0 , then it unfolds uniquely to a (TE)-structure with primitive Higgs field over
34 C. Hertling
a germ of a regular F -manifold by Malgrange’s result Theorem 3.16(c). The germ of the regular
F -manifold is uniquely determined by the isomorphism class of U : Kt0 → Kt0 (i.e., its Jordan
block structure). And the (TE)-structure is uniquely determined by its restriction to t0.
The following statement on the rank 2 cases is an immediate consequence of Malgrange’s un-
folding result Theorem 3.16(c), the classification of germs of regular 2-dimensional F -manifolds
in Remark 2.6(ii) (building on Theorems 2.2 and 2.3, see also Remark 3.17(iii)) and the clas-
sification of the rank 2 (TE)-structures into the cases (Sem), (Bra), (Reg) and (Log) in Defini-
tion 4.4.
Corollary 5.1.
(a) For any rank 2 (TE)-structure over a point t0 except those of type (Log), the endomorphism
U : Kt0 → Kt0 is regular. The (TE)-structure has a unique universal unfolding. This
unfolding has a primitive Higgs field. Its base space is a germ
(
M, t0
)
=
(
C2, 0
)
of an
F -manifold with Euler field and is as follows:
Type F -manifold Euler field
(Sem) A2
1
∑2
i=1(ui + ci)ei with c1 ̸= c2
(Bra) or (Reg) N2 t1∂1 + g(t2)∂2 with g(0) ̸= 0
In the case of (Bra) or (Reg), a coordinate change brings E to the form t1∂1 + ∂2.
(b) Any unfolding of a rank 2 (TE)-structure over t0 with regular endomorphism U : Kt0 → Kt0
is induced by the universal unfolding in (a).
Because of the existence and uniqueness of the universal unfolding, it is not really necessary
to give it explicitly. On the other hand, in rank 2, it is easy to give it explicitly. The following
lemma offers one way.
Lemma 5.2. Let (H → C,∇) by a (TE)-structure over a point with monodromy Mmon of
some rank r ∈ N. It has an unfolding which is a (TE)-structure
(
H(unf) → C ×M,∇
)
, where
M = C× C∗ with coordinates t = (t1, t2) (on C2 ⊃M), with the following properties.
(a) The monodromy around t2 = 0 is (Mmon)−1.
(b) The original (TE)-structure is isomorphic to the one over t0 = (0, 1).
(c) If v0 is a C{z}-basis of O(H)0 with z2∇∂zv
0 = v0B0, then H(unf) has over (C, 0) ×M
a basis v such that the matrices A1, A2 and B in (3.2)–(3.4) are as follows
A1 = C1, (5.1)
A2 = −B0
(
z
t2
)
, (5.2)
B = −t1C1 + t2B
0
(
z
t2
)
= −t1A1 − t2A2. (5.3)
(d) If U|t0 is regular and rankH = 2, then the Higgs field of the (TE)-structure H(unf) is
everywhere primitive. Therefore then M is an F -manifold with Euler field. The Euler
field is E = t1∂1 + t2∂2.
(e) If U|t0 is regular and rankH = 2, the (TE)-structure over the germ
(
M, t0
)
is the universal
unfolding of the one over t0.
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 35
Proof. Let f0 =
(
f01 , . . . , f
0
r
)
be a flat multivalued basis of H ′ := H|C∗ . Let Mmat ∈ GLr(C)
be the matrix of its monodromy, so f0
(
ze2πi
)
= f0 ·Mmat. Let v0 =
(
v01, . . . , v
0
r
)
be a C{z}-
basis of O(H)0. Let B0 ∈ GLr(C{z}) be the matrix with z2∇∂zv
0 = v0B0. Consider the
matrix Ψ(z, t) with multivalued entries with
v0 = f0 ·Ψ(z).
Then
Ψ
(
ze2πi
)
=
(
Mmat
)−1 ·Ψ(z),
Ψ−1∂zΨ = z−2B0(z).
Embed the flat bundle H ′ := H|C∗ as the bundle over t0 = (0, 1) into a flat bundle H(mf)′ →
C∗ × M with monodromy Mmon around z = 0 and monodromy (Mmon)−1 around t2 = 0.
The flat multivalued basis f0 of H ′ extends to a flat multivalued basis f of H(mf)′ with
f
(
ze2πi, t
)
= f(z, t)Mmat,
f
(
z, t1, t2e
2πi
)
= f(z, t)
(
Mmat
)−1
.
The tuple of sections v with
v = f · et1/zΨ
(
z
t2
)
is univalued, it is a basis of H(mf)′ in a neighbourhood of {0} ×M , and it has the connection
matrices in (5.1)–(5.3): The calculations for A2 and B are
∇∂2v = f et1/z
(
− z
t22
)
(∂zΨ)
(
z
t2
)
= f et1/z
(
− z
t22
)
Ψ
(
z
t2
)(
z
t2
)−2
B0
(
z
t2
)
= v
(
−1
z
)
B0
(
z
t2
)
,
∇∂zv = f et1/z
((
− t1
z2
)
Ψ
(
z
t2
)
+
(
1
t2
)
(∂zΨ)
(
z
t2
))
= f et1/z
((
− t1
z2
)
Ψ
(
z
t2
)
+
(
1
t2
)
Ψ
(
z
t2
)(
z
t2
)−2
B0
(
z
t2
))
= v
((
− t1
z2
)
C1 +
(
t2
z2
)
B0
(
z
t2
))
.
Therefore v defines a (TE)-structure, which we call (H(unf) → C×M,∇). It unfolds the one
over t0 = (0, 1), and that one is isomorphic to (H → C,∇).
It rests to show (d) and (e). Suppose rankH = 2. Then U|t0 is regular if and only if(
B0
)(0)
/∈ C · C1. Then also A
(0)
2 (t) = −
(
B0
)(0)
/∈ C · C1, so then the Higgs field of the (TE)-
structure H(unf) is everywhere primitive. Because of B(0) = −t1A(0)
1 − t2A
(0)
2 , the Euler field is
E = t1∂1 + t2∂2. (e) follows from (d) and Malgrange’s result Theorem 3.16(c). ■
Remarks 5.3.
(i) In the cases (Reg) we will see the universal unfoldings again in Section 7, in Remarks 7.2.
In a first step in Remarks 7.1, the value t2 in the normal form in Remarks 4.19 is turned
into a parameter in P1. Remarks 7.2 add another parameter t1 in C. Then the Higgs field
becomes primitive and the base space C × P1 becomes a 2-dimensional F -manifold with
Euler field. For each t0 ∈ C × C∗, the (TE)-structure over t0 is of type (Reg), and the
(TE)-structure over the germ
(
M, t0
)
is a universal unfolding of the one over t0.
36 C. Hertling
(ii) In the cases (Bra), the following formulas give a universal unfolding over
(
C2, 0
)
of any
(TE)-structure of type (Bra) over the point 0 (see Theorem 4.11 for their classification),
such that the Euler field is E = (t1 + c1)∂1 + ∂2. Here ρ
(1) ∈ C, b(0)3 ∈ C, b(0)2 , b
(1)
4 ∈ C∗,
A1 = C1,
A2 = −b(0)2 C2 − z
(
1
2
+ b
(1)
3
)
D − zb
(1)
4 et2E,
B = (−t1 − c1)C1 + b
(0)
2 C2 + z
(
ρ(1)C1 + b
(1)
3 D + b
(1)
4 et2E
)
= (−t1 − c1)A1 −A2 + zρ(1)C1 − z
1
2
D.
(iii) In the cases (Sem), a (TE)-structure over a point extends uniquely to a (TE)-structure
over the universal covering M of the manifold
{
(u1, u2) ∈ C2 |u1 ̸= u2
}
(see [17] and [21,
Chapter III, Theorem 2.10]). For each t0 ∈ M the (TE)-structure over t0 is of type
(Sem), and the (TE)-structure over the germ
(
M, t0
)
is the universal unfolding of the
(TE)-structure over t0.
6 1-parameter unfoldings of logarithmic (TE)-structures
over a point
This section classifies unfoldings over
(
M, t0
)
= (C, 0) with trace free pole part of logarithmic
(TE)-structures over the point t0.
It is a preparation for Section 8, which treats arbitrary unfoldings of (TE)-structures of type
(Log) over a point.
Section 6.1: An unfolding with trace free pole part over
(
M, t0
)
= (C, 0) of a logarithmic
rank 2 (TE)-structure over t0 will be considered. Invariants of it will be defined. Theorem 6.2
gives constraints on these invariants and shows that the monodromy is semisimple if the generic
type is (Sem) or (Bra).
By Theorem 3.20(a) (which is trivial in our case because of the logarithmic pole at z = 0
of the (TE)-structure over t0) and Remark 3.19(iii), the (TE)-structure has a Birkhoff normal
form, i.e., an extension to a pure (TLE)-structure, if its monodromy is semisimple.
Section 6.2: All pure (TLE)-structures over
(
M, t0
)
= (C, 0) with trace free pole part and
with logarithmic restriction to t0 are classified in Theorem 6.3. These comprise all with semisim-
ple monodromy and thus all with generic types (Sem) or (Bra).
Section 6.3: All (TE)-structures over
(
M, t0
)
= (C, 0) with trace free pole part and with
logarithmic restriction over t0 whose monodromies have a 2 × 2 Jordan block are classified in
Theorem 6.7. Their generic types are (Reg) or (Log) because of Theorem 6.2. Most of them
have no Birkhoff normal forms. The intersection with Theorem 6.3 is small and consists of those
which have Birkhoff normal forms.
Theorems 6.3 and 6.7 together give all unfoldings with trace free pole parts over
(
M, t0
)
=
(C, 0) of logarithmic rank 2 (TE)-structures over t0.
6.1 Numerical invariants for such (TE)-structures
The next definition gives some numerical invariants for such (TE)-structures. Recall the invari-
ants δ(0) and δ(1) in Lemma 3.9.
Definition 6.1. Let
(
H → C×
(
M, t0
)
,∇
)
be a (TE)-structure with trace free pole part over(
M, t0
)
= (C, 0) (with coordinate t) whose restriction over t0 = 0 is logarithmic. Let M ⊂ C
be a neighborhood of 0 on which the (TE)-structure is defined. On M \ {0} it has a fixed
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 37
type, (Sem) or (Bra) or (Reg) or (Log), which is called the generic type of the (TE)-structure.
Lemma 4.3 characterizes the generic type in terms of (non)vanishing of δ(0), δ(1) ∈ tC{t} and U :
(Sem) (Bra) (Reg) (Log)
δ(0) ̸= 0 δ(0) = 0, δ(1) ̸= 0 δ(0) = δ(1) = 0, U ̸= 0 U = 0
For the generic types (Sem), (Bra) and (Reg), define k1 ∈ N by
k1 := max(k ∈ N | U(O(H)0) ⊂ tkO(H)0. (6.1)
For the generic types (Sem) and (Bra) define k2 ∈ Z by
k2 :=
{
degt δ
(0) − k1 for the generic type (Sem),
degt δ
(1) − k1 for the generic type (Bra).
The following theorem gives for the generic type (Bra) and part of the generic type (Sem)
restrictions on the eigenvalues of the residue endomorphism of the logarithmic pole at z = 0 of
the (TE)-structure over t0 = 0. And it shows that the monodromy is semisimple if the generic
type is (Sem) or (Bra).
Theorem 6.2. Let
(
H → C×
(
M, t0
)
,∇
)
be a rank 2 (TE)-structure with trace free pole part
over
(
M, t0
)
= (C, 0) whose restriction over t0 = 0 is logarithmic. Recall the invariant ρ(1) ∈ C
from Lemma 3.9(b), and recall the invariants k1 ∈ N and k2 ∈ Z from Definition 6.1 if the
generic type is (Sem) or (Bra).
(a) Suppose that the generic type is (Sem).
(i) Then k2 ≥ k1.
(ii) If k2 > k1 then the eigenvalues of the residue endomorphism of the logarithmic pole
at z = 0 of the (TE)-structure over t0 are ρ(1) ± k1−k2
2(k1+k2)
. Their difference is smaller
than 1. Especially, the eigenvalues of the monodromy are different, and the mono-
dromy is semisimple.
(iii) Also if k1 = k2, the monodromy is semisimple.
(b) Suppose that the generic type is (Bra).
(i) Then k2 ∈ N.
(ii) The eigenvalues of the residue endomorphism of the logarithmic pole at z = 0 of
the (TE)-structure over t0 are ρ(1) ± −k2
2(k1+k2)
. Their difference is smaller than 1.
Especially, the eigenvalues of the monodromy are different, and the monodromy is
semisimple.
Proof. By Lemma 3.11, a C{t, z}-basis v of the germ O(H)(0,0) can be chosen such that the
matrices A and B ∈ M2×2(C{t, z}) with z∇∂tv = vA and z2∇∂zv = vB satisfy (3.14), 0 =
trA = tr
(
B − zρ(1)C1
)
, or, more explicitly,
A = a2C2 + a3D + a4E with a2, a3, a4 ∈ C{t, z},
B = zρ(1)C1 + b2C2 + b3D + b4E with b2, b3, b4 ∈ C{t, z}.
Write aj =
∑
k≥0 a
(k)
j zk and a
(k)
j =
∑
l≥0 a
(k)
j,l t
l ∈ C{t}, and analogously for bj . Condition (3.8)
says here
0 = z∂tB − z2∂zA+ zA+ [A,B]
38 C. Hertling
= C2
[
z∂tb2 + za
(0)
2 −
∑
k≥2
(k − 1)a
(k)
2 zk+1 + 2a2b3 − 2a3b2
]
(6.2)
+D
[
z∂tb3 + za
(0)
3 −
∑
k≥2
(k − 1)a
(k)
3 zk+1 − a2b4 + a4b2
]
(6.3)
+ E
[
z∂tb4 + za
(0)
4 −
∑
k≥2
(k − 1)a
(k)
4 zk+1 − 2a4b3 + 2a3b4
]
. (6.4)
(a) Suppose that the generic type is (Sem).
(i) By definition of k1 and k2,
k1 = min
(
degt b
(0)
2 , degt b
(0)
3 ,degt b
(0)
4
)
, (6.5)
k1 + k2 = degt
((
b
(0)
3
)2
+ b
(0)
2 b
(0)
4
)
≥ 2k1, (6.6)
thus k2 ≥ k1.
(ii) Suppose k2 > k1. By a linear change of the basis v, we can arrange that k1 = degt b
(0)
2 .
The base change matrix T = C1 + b
(0)
3 /b
(0)
2 · E ∈ GL2(C{t}) gives the new basis ṽ = v · T with
matrix
B̃(0) = T−1B(0)T = b
(0)
2 C2 +
(
b
(0)
4 +
(
b
(0)
3
)2
b
(0)
2
)
E.
We can make a coordinate change in t such that afterwards
b
(0)
2 b
(0)
4 +
(
b
(0)
3
)2
= γ2tk1+k2
for an arbitrarily chosen γ ∈ C∗. Then a diagonal base change leads to a basis which is again
called v with matrices which are again called A and B with
b
(0)
3 = 0, b
(0)
2 = γtk1 , b
(0)
4 = γtk2 .
Now the vanishing of the coefficients in C{t} of C2 ·z0, C2 ·z1, D ·z0, D ·z1 and E ·z1 in (6.2)–(6.4)
tells the following:
C2 · z0 : a
(0)
3 = 0,
C2 · z1 : 0 = k1γt
k1−1 + a
(0)
2
(
1 + 2b
(1)
3
)
− 2a
(1)
3 γtk1 ,
so degt a
(0)
2 = k1 − 1, 0 = k1γ + a
(0)
2,k1−1
(
1 + 2b
(1)
3,0
)
,
D · z0 : a
(0)
2 γtk2 = a
(0)
4 γtk1 , so a
(0)
4 = a
(0)
2 tk2−k1 ,
so deg a
(0)
4 = k2 − 1, and a
(0)
4,k2−1 = a
(0)
2,k1−1.
D · z1 : a
(0)
2 b
(1)
4 + a
(1)
2 γtk2 = a
(0)
4 b
(1)
2 + a
(1)
4 γtk1 ,
so b
(1)
4,0 = 0 (here k2 > k1 is used),
E · z1 : 0 = k2γt
k2−1 + a
(0)
4
(
1− 2b
(1)
3
)
+ 2a
(1)
3 γtk2 ,
so 0 = k2γ + a
(0)
4,k2−1
(
1− 2b
(1)
3,0
)
.
This shows
b
(1)
4,0 = 0, b
(1)
3,0 =
k1 − k2
2(k1 + k2)
∈
(
−1
2
, 0
)
∩Q.
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 39
With respect to the basis v|(0,0) of K(0,0), the matrix of the residue endomorphism of the loga-
rithmic pole at z = 0 of the (TE)-structure over t0 = 0 is
B(1)(0) = ρ(1)C1 + b
(1)
3,0D + b
(1)
2,0C2.
It is semisimple with the eigenvalues ρ(1) ± b
(1)
3,0, whose difference is smaller than 1. The mon-
odromy is semisimple with the two different eigenvalues exp
(
−2πi
(
ρ(1) ± b
(1)
3,0
))
.
(iii) Suppose k2 = k1. As in the proof of (ii), we can make a coordinate change in t and then
obtain a C{t, z}-basis ṽ of O(H)(0,0) with
b̃
(0)
3 = 0, b̃
(0)
2 = b̃
(0)
4 = γtk1
for an arbitrarily chosen γ ∈ C∗. Now the constant base change matrix T =
(
1 1
1 −1
)
gives the
basis v = ṽ · T with
b
(0)
2 = b
(0)
4 = 0, b
(0)
3 = γtk1 .
The vanishing of the coefficients in C{t} of C2 · z0, E · z0, D · z1, C2 · z1 and E · z1 in (6.2)–(6.4)
tells the following:
C2 · z0 : a
(0)
2 = 0,
E · z0 : a
(0)
4 = 0,
D · z1 : 0 = k1γt
k1−1 + a
(0)
3 , so a
(0)
3 = −k1γtk1−1,
C2 · z1 : b
(1)
2 =
b
(0)
3
a
(0)
3
a
(1)
2 =
−1
k1
· t · a(1)2 , so b
(1)
2,0 = 0,
E · z1 : b
(1)
4 =
b
(0)
3
a
(0)
3
a
(1)
4 =
−1
k1
· t · a(1)4 , so b
(1)
4,0 = 0.
With respect to the basis v|(0,0) of K(0,0), the matrix of the residue endomorphism of the loga-
rithmic pole at z = 0 of the (TE)-structure over t0 = 0 is
B(1)(0) = ρ(1)C1 + b
(1)
3,0D.
It is diagonal with the eigenvalues ρ(1) ± b
(1)
3,0. Therefore the monodromy has the eigenvalues
exp
(
−2πi
(
ρ(1) ± b
(1)
3,0
))
.
If b
(1)
3,0 ∈ C\
(
1
2Z\{0}
)
, the eigenvalues of the residue endomorphism do not differ by a nonzero
integer. Because of Theorem 3.23(c), then the monodromy is semisimple.
We will show that the monodromy is also in the cases b
(1)
3,0 ∈ 1
2Z\{0} semisimple, by reducing
these cases to the case b
(1)
3,0 = 0.
Suppose b
(1)
3,0 ∈ 1
2N. The case b
(1)
3,0 ∈ 1
2Z<0 can be reduced to this case by exchanging v1 and v2.
We will construct a new (TE)-structure over
(
M, t0
)
= (C, 0) with the same monodromy and
again with trace free pole part and of generic type (Sem) with logarithmic restriction over t0,
but where B(1)(0) is replaced by
B̃(1)(0) =
(
ρ(1) +
1
2
)
+
(
b
(1)
3,0 −
1
2
)
D.
Applying this sufficiently often, we arrive at the case b
(1)
3,0 = 0, which has semisimple monodromy.
40 C. Hertling
The basis ṽ := v · ( 1 0
0 z ) of H ′ := H|C∗×(M,t0) in a neighborhood of (0, 0) defines a new
(TE)-structure over (M, 0) because of
z∇∂t ṽ = ṽ
(
z−1a2C2 + a3D + za4E
)
and a
(0)
2 = 0,
z2∇∂z ṽ = ṽ
(
z
(
ρ(1) +
1
2
)
C1 + z−1b2C2 +
(
b3 − z
1
2
)
D + zb4E
)
and b
(0)
2 = 0.
Of course, it has the same monodromy. The restriction over t0 = 0 has a logarithmic pole at
z = 0 because b
(1)
2 = −1
k1
ta
(1)
2 and b
(0)
3 = γtk1 with k1 ∈ N. Its generic type is still (Sem). Its
numbers k̃1 and k̃2 satisfy k̃1 + k̃2 = degt det Ũ = degt
(
b
(0)
3
)2
= 2k1. The assumption k̃1 < k̃2
would lead together with part (ii) to two different eigenvalues of the monodromy, a contradiction.
Therefore k̃1 = k̃2 = k1. Thus we are in the same situation as before, with b
(1)
3,0 diminuished by 1
2 .
(b) Suppose that the generic type is (Bra).
(i) and (ii) U is nilpotent, but not 0. We can choose a C{t, z}-basis v of O(H)(0,0) such that
B(0) = b
(0)
2 C2, so b
(0)
3 = b
(0)
4 = 0.
Then δ(1) = −b(0)2 b
(1)
4 . Here degt b
(0)
2 = k1 and degt δ
(1) = k1 + k2, so k2 = degt b
(1)
4 ≥ 0. We can
make a coordinate change in t such that afterwards
b
(0)
2 b
(1)
4 = γ2tk1+k2 ,
for an arbitrarily chosen γ ∈ C∗. Then a diagonal base change leads to a basis which is again
called v with matrices which are again called A and B with
b
(0)
2 = γtk1 , b
(0)
3 = b
(0)
4 = 0, b
(1)
4 = γtk2 .
The vanishing of the coefficients in C{t} of C2 · z0, D · z0, C2 · z1, D · z1 and E · z2 in (6.2)–(6.4)
tells the following
C2 · z0 : a
(0)
3 = 0,
D · z0 : a
(0)
4 = 0,
C2 · z1 : 0 = k1γt
k1−1 + a
(0)
2
(
1 + 2b
(1)
3
)
− 2a
(1)
3 γtk1 ,
so degt a
(0)
2 = k1 − 1, 0 = k1γ + a
(0)
2,k1−1
(
1 + 2b
(1)
3,0
)
,
D · z1 : a
(0)
2 γtk2 = a
(1)
4 γtk1 , so tk2 = a
(1)
4
tk1
a
(0)
2
,
so k2 = 1 + deg a
(1)
4 ≥ 1, and a
(1)
4 = a
(0)
2 tk2−k1 ,
E · z2 : 0 = k2γt
k2−1 + 2a
(1)
3 γtk2 − 2a
(1)
4 b
(1)
3 ,
so 0 = k2γ − 2a
(0)
2,k1−1b
(1)
3,0.
This shows
k2 ≥ 1, b
(1)
4,0 = 0, b
(1)
3,0 =
−k2
2(k1 + k2)
∈
(
−1
2
, 0
)
∩Q.
With respect to the basis v|(0,0) of K(0,0), the matrix of the residue endomorphism of the loga-
rithmic pole at z = 0 of the (TE)-structure over t0 = 0 is
B(1)(0) = ρ(1)C1 + b
(1)
3,0D + b
(1)
2,0C2.
It is semisimple with the eigenvalues ρ(1) ± b
(1)
3,0, whose difference is smaller than 1. The mono-
dromy is semisimple with the two different eigenvalues exp
(
−2πi
(
ρ(1) ± b
(1)
3,0
))
. ■
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 41
6.2 1-parameter unfoldings with trace free pole part of logarithmic
pure (TLE)-structures over a point
Such unfoldings are themselves pure (TLE)-structures over (C, 0), see Remark 3.19(iii) respec-
tively [21, Chapter VI, Theorem 2.1] or [7, Theorem 5.1(c)]. Their restrictions over t0 = 0 have
a logarithmic pole at z = 0. Theorem 6.3 classifies such pure (TLE)-structures. The underly-
ing (TE)-structures were subject of Definition 6.1 and Theorem 6.2. They gave their generic
type and invariants (k1, k2) ∈ N2 (for the generic types (Sem) and (Bra)) and k1 ∈ N (for the
generic type (Reg)). Theorem 6.3 will give an invariant k1 ∈ N also for the generic type (Log)
with Higgs field ̸= 0. Lemma 3.9(b) gave the invariant ρ(1) ∈ C. The coordinate on C is again
called t.
Theorem 6.3. Any pure rank 2 (TLE)-structure over
(
M, t0
)
= (C, 0) with trace free pole part
and with logarithmic restriction over t0 has after a suitable coordinate change in t a unique
Birkhoff normal form in the following list. Here the Birkhoff normal form consists of two
matrices A and B which are associated to a global basis v of H whose restriction to {∞} ×(
M, t0
)
is flat with respect to the residual connection along {∞}×
(
M, t0
)
, via z∇∂tv = vA and
z2∇∂zv = vB. The matrices have the shape
A = a
(0)
2 C2 + a
(0)
3 D + a
(0)
4 E,
B = zρ(1)C1 − γtA+ zb
(1)
2 C2 + zb
(1)
3 D, (6.7)
with a
(0)
2 , a
(0)
3 , a
(0)
4 ∈ C[t], ρ(1), γ ∈ C, b(1)2 , b
(1)
3 ∈ C (so here zb
(1)
4 E does not turn up, resp. b
(1)
4
= 0). The left column of the following list gives the generic type of the underlying (TE)-structure
and, depending on the type, the invariant k1 ∈ N or the invariants k1, k2 ∈ N from Definition 6.1
of the underlying (TE)-structure. The invariant ρ(1) ∈ C is arbitrary and is not listed in the
table. ζ ∈ C, α3 ∈ R≥0 ∪ H, α4 ∈ C \ {−1}, k1 ∈ N and k2 ∈ N are invariants in some cases.
In the first 6 cases, a
(0)
i is determined by b
(0)
i = −γta(0)i .
Generic type
and invariants γ b
(0)
2 b
(0)
3 b
(0)
4 b
(1)
2 b
(1)
3
(Sem)
k2 − k1 > 0 odd 2
k1+k2
tk1 0 tk2 0 k1−k2
2(k1+k2)
k2 − k1 ∈ 2N 2
k1+k2
tk1 ζt(k1+k2)/2
(
1− ζ2
)
tk2 0 k1−k2
2(k1+k2)
k2 = k1
1
k1
0 tk1 0 0 α3
(Bra), k1, k2
1
k1+k2
tk1 tk1+k2 −tk1+2k2 0 −k2
2(k1+k2)
(Reg), k1
1+α4
k1
tk1 0 0 0 1
2α4
(Reg), k1
1
k1
tk1 0 0 1 0
Generic type γ a
(0)
2 a
(0)
3 a
(0)
4 b
(1)
2 b
(1)
3
(Log) 0 k1t
k1−1 0 0 0 −1
2
(Log) 0 0 0 0 0 α3
(Log) 0 0 0 0 1 0
Before the proof, several remarks on these Birkhoff normal forms are made. The proof is
given after Remark 6.6.
42 C. Hertling
Remarks 6.4.
(i) The matrix B(0) = zB(1)(0) is the matrix of the logarithmic pole at z = 0 of the restriction
over t0 = 0 of the (TE)-structure. In all cases except the 6th case and the 9th case,
it is z
(
ρ(1)C1 + b
(1)
3 D
)
, so it is diagonal. In these cases the monodromy is semisimple
with eigenvalues exp
(
−2πi
(
ρ(1) ± b
(1)
3
))
. In the 6th case and the 9th case, this matrix
is z
(
ρ(1)C1 + C2
)
. Then the matrix and the monodromy have a 2 × 2 Jordan block, and
the monodromy has the eigenvalue exp
(
−2πiρ(1)
)
. In all cases, the leading exponents
(defined in Theorem 4.20) of the logarithmic (TE)-structure over t0 are called α0
1 and α0
2,
and they are
α0
1/2 = ρ(1) ± b
(1)
3 , i.e.,
α0
1 + α0
2
2
= ρ(1), α0
1 − α0
2 = 2b
(1)
3 .
The 6th and 9th cases turn up again in Theorem 6.7. See Remarks 6.8(iv)–(vi).
(ii) In the generic types (Sem), the critical values satisfy u2 = −u1 because the pole part is
trace free, −u1+u2
2 = ρ(0) = 0. They and the regular singular exponents α1 and α2 can be
calculated with the formulas (4.1) and (4.2):
δ(0) = −b(0)2 b
(0)
4 −
(
b
(0)
3
)2
= −tk1+k2 , (6.8)
u1/2 = ±
√
1
4
(u1 − u2)2 = ±
√
−δ(0) = ±t(k1+k2)/2, (6.9)
α1 + α2
2
= ρ(1), (6.10)
α1 − α2 = u−1
1 δ(1) =
0, gen. type (Sem) with k2 − k1 > 0 odd,
k2−k1
k1+k2
ζ, gen. type (Sem) with k2 − k1 ∈ 2N,
−2α3, gen. type (Sem) with k2 = k1.
(6.11)
If k2 = k1 then {α1, α2} =
{
α0
1, α
0
2
}
, but if k2 > k1 then {α1, α2} ̸=
{
α0
1, α
0
2
}
, except if
ζ ∈ {±1}.
(iii) In the generic type (Bra), ρ(1) ∈ C is arbitrary, b
(1)
3 = −k2
2(k1+k2)
, and δ(1) varies as follows,
δ(1) =
k2
k1 + k2
tk1+k2 . (6.12)
(iv) In the 5th, 7th and 8th cases in Theorem 6.3, the monodromy is semisimple and the
(TE)-structure is regular singular. Associate to it the data in Definition 3.18: H ′ :=
H|C×(M,t0), M
mon, Nmon, Eig(Mmon) = {λ1, λ2}, H∞, Cα for α ∈ C with e−2πiαj ∈
{λ1, λ2}. The leading exponents of the logarithmic (TE)-structure over t0 are called α0
1
and α0
2 as in (i). The leading exponents of the (TE)-structure over t ∈ C \ {0} are now
called α1 and α2. Possibly after renumbering λ1 and λ2, α
0
1 and α0
2, and α1 and α2, we
have e−2πiα0
j = e−2πiαj = λj and the relations in the following table:
In Theorem 6.3 α0
1 α0
2 α1 α2
5th case ρ(1) + 1
2α4 ρ(1) − 1
2α4 α0
1 α0
2 − 1
7th case ρ(1) − 1
2 ρ(1) + 1
2 α0
1 α0
2
8th case ρ(1) + α3 ρ(1) − α3 α0
1 α0
2
(6.13)
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 43
And there exist sections sj ∈ Cαj \ {0} with
O(H)0 = C{t, z}
(
s1 +
−1
1 + α4
tk1s2
)
⊕ C{t, z}(zs2) in the 5th case, (6.14)
O(H)0 = C{t, z}
(
s1 + tk1z−1s2
)
⊕ C{t, z}s2 in the 7th case, (6.15)
O(H)0 = C{t, z}s1 ⊕ C{t, z}s2 in the 8th case. (6.16)
One confirms (6.14)–(6.16) immediately by calculating the matrices A and B with
z∇∂tv = vA and z2∇∂zv = vB for v the basis in (6.14)–(6.16).
(v) Theorem 6.7 contains for the 6th and 9th cases in Theorem 6.3 a description similar to
part (iv). See Remarks 6.8(iv)–(vi).
Remarks 6.5. These remarks study the behaviour of the (TE)-structures in Theorem 6.3 under
pull back via maps φ : (C, 0) → (C, 0). The normal forms in Theorem 6.3 are chosen such that
the pull backs by maps φ with φ(s) = sn for some n ∈ N are again normal forms in Theorem 6.3.
(i) A general observation: Let
(
H → C×
(
M, t0
)
,∇
)
be a (TE)-structure over
(
M, t0
)
= (C, 0)
of rank r ∈ N. Let v be C{t, z}-basis of O(H)0 with z∇∂tv = vA and z2∇∂zv = vB
and A,B ∈ Mr×r(C{t, z}). Choose n ∈ N and consider a map φ : (C, 0) → (C, 0), s 7→
φ(s) = t. Then the pull back (TE)-structure φ∗(H,∇) has the basis ṽ(z, s) = v(z, φ(s)).
The matrices Ã, B̃ ∈Mr×r(C{s, z}) with z∇∂s ṽ = ṽÃ, z2∇∂z ṽ = ṽB̃ are
à = ∂sφ(s) ·A(z, φ(s)), B̃ = B(z, φ(s)). (6.17)
(ii) These formulas (6.17) show for the 1st to 7th cases in the list in Theorem 6.3 the following:
The pull back via φ : (C, 0) → (C, 0) with φ(s) = sn for some n ∈ N of such a (TE)-
structure with invariants (k1, k2) or k1 is a (TE)-structure in the same case, where the
invariants (k1, k2) or k1 are replaced by
(
k̃1, k̃2
)
= (nk1, nk2) or k̃1 = nk1, and where all
other invariants coincide with the old invariants.
(iii) The following table says which of the (TE)-structures in the 1st to 7th cases in the list in
Theorem 6.3 are not induced by other such (TE)-structures:
Generic type and invariants Not induced if
(Sem): k2 − k1 > 0 odd gcd(k1, k2) = 1
(Sem): k2 − k1 ∈ 2N, ζ = 0 gcd(k1, k2) = 1
(Sem): k2 − k1 ∈ 2N, ζ ̸= 0 gcd
(
k1,
k1+k2
2
)
= 1
(Sem): k2 = k1 k2 = k1 = 1
(Bra) gcd(k1, k2) = 1
(Reg) : Nmon = 0 k1 = 1
(Reg) : Nmon ̸= 0 k1 = 1
(Log) k1 = 1
(iv) In the 8th and 9th cases, the (TE)-structure is induced by its restriction over t0 via the
map φ :
(
M, t0
)
→
{
t0
}
, so it is constant along M .
(v) The formulas (6.14) and (6.15) confirm part (ii) for the 5th and 7th cases in Theorem 6.3.
Formula (6.16) confirms part (iv) in the 8th case in Theorem 6.3. Analogous statements
to part (ii) and part (iv) hold for the cases in Theorem 6.7. They follow from the formu-
las (6.26), (6.27) and (6.28) there, which are analogous to (6.14), (6.15) and (6.16). See
Remarks 6.8(ii) and (iii).
44 C. Hertling
Remark 6.6. In the 2nd and 4th cases in the list in Theorem 6.3, another C{t, z}-basis ṽ
of O(H)0 with nice matrices à and B̃ is
ṽ = v · T with T = C1 +
a
(0)
3
a
(0)
2
E =
{
C1 + ζt(k2−k1)/2E in the 2nd case,
C1 + tk2E in the 4th case.
In the 2nd case
à = −γ−1
(
tk1−1C2 + tk2−1E
)
+ z
k2 − k1
2
ζt(k2−k1−2)/2E,
B̃ = zρ(1)C1 − γtÃ+ zb
(1)
3 D.
In the 4th case
à = −γ−1tk1−1C2 + zk2t
k2−1E,
B̃ = zρ(1)C1 − γtÃ+ zb
(1)
3 D.
These matrices are not in Birkhoff normal form. The basis ṽ is still a global basis of the pure
(TLE)-structure, but the sections ṽj |{∞}×M are not flat with respect to the residual connection
along {∞} ×M .
Proof of Theorem 6.3. Consider any pure (TLE)-structure over
(
M, t0
)
= (C, 0) with trace
free pole part and with logarithmic restriction to t0. Choose a global basis v of H whose
restriction to {∞}×
(
M, t0
)
is flat with respect to the residual connection along {∞}×
(
M, t0
)
.
Its matrices A and B with z∇∂tv = vA and z2∇∂zv = vB have because of (3.13) (in Lemma 3.11)
the shape (6.7) and
B = zρ(1)C1 +
(
b
(0)
2 + zb
(1)
2
)
C2 +
(
b
(0)
3 + zb
(1)
3
)
D +
(
b
(0)
4 + zb
(1)
4
)
E
with a
(0)
j ∈ C{t}, b(0)j ∈ tC{t}, b(1)j ∈ C. They satisfy the relations (3.19) (and, equiva-
lently, (6.2)–(6.4)), so, more explicitly,
a
(0)
i b
(0)
j = a
(0)
j b
(0)
i for (i, j) ∈ {(2, 3), (2, 4), (3, 4)}, (6.18)−∂tb(0)2
−∂tb(0)3
−∂tb(0)4
=
1 + 2b
(1)
3 −2b
(1)
2 0
−b(1)4 1 b
(1)
2
0 2b
(1)
4 1− 2b
(1)
3
a
(0)
2
a
(0)
3
a
(0)
4
. (6.19)
First we consider the cases when all a
(0)
j are 0. Then also all b
(0)
j are 0, because of b
(0)
j ∈ tC{t}
and because of the differential equations (6.19). Then B = zB(1), and it is clear that this matrix
can be brought to the form B = zρ(1)C1 + zα3D or B = zρ(1)C1 + zC2 by a constant base
change. The number α3 ∈ C can be replaced by −α3, so α3 ∈ R≥0 ∪H is unique. This gives the
last two cases in the list. There the generic type is (Log).
For the rest of the proof, we consider the cases when at least one a
(0)
j is not 0. Then (6.18)
says
(
b
(0)
2 , b
(0)
3 , b
(0)
4
)
=
b
(0)
j
a
(0)
j
·
(
a
(0)
2 , a
(0)
3 , a
(0)
4
)
, so B(0) =
b
(0)
j
a
(0)
j
·A(0). (6.20)
If b
(0)
j = 0 then b
(0)
2 = b
(0)
3 = b
(0)
4 = 0, and the generic type is (Log). If b
(0)
j ̸= 0, then the
generic type is (Sem) or (Bra) or (Reg). Consider for a moment the cases when the residue
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 45
endomorphism of the logarithmic pole at z = 0 of the (TE)-structure over t0 is semisimple.
By Theorem 6.1, these cases include the generic types (Sem) and (Bra). Then a linear base
change gives b
(1)
2 = b
(1)
4 = 0, so that the 3 × 3-matrix in (6.19) is diagonal. Then denote
β̃j := degt b
(0)
j ∈ N. A coordinate change in t leads to b
(0)
j = b
(0)
j,β̃j
· tβ̃j . The differential equation
in (6.19) leads to a
(0)
j = a
(0)
j,β̃j−1
· tβ̃j−1, and to b
(0)
j /a
(0)
j = −γt for some γ ∈ C∗. Define
β2 =
1 + 2b
(1)
3
γ
, β3 =
1
γ
, β4 =
1− 2b
(1)
3
γ
. (6.21)
Now (6.20) and the differential equations in (6.19) show β̃j = βj and
b
(0)
2 = 0 or
(
β2 ∈ N and b
(0)
2 = b
(0)
2,β2
· tβ2
)
̸= 0,
b
(0)
3 = 0 or
(
β3 ∈ N and b
(0)
3 = b
(0)
3,β3
· tβ3
)
̸= 0, (6.22)
b
(0)
4 = 0 or
(
β4 ∈ N and b
(0)
4 = b
(0)
4,β4
· tβ4
)
̸= 0.
Now we discuss the generic types (Sem), (Bra), (Reg) and (Log) separately.
Generic type (Sem). By Theorem 6.2, we can choose the basis v such that b
(1)
2 = b
(1)
4 = 0. In the
cases k2 > k1, by Theorem 6.2, b
(1)
3 is up to the sign unique, and we can choose it to be
b
(1)
3 =
k1 − k2
2(k1 + k2)
∈
(
−1
2
, 0
)
∩Q
(possibly by exchanging v1 and v2). In the cases k2 = k1 we write α3 := b
(1)
3 ∈ C. We can
change its sign and get a unique α3 ∈ R≥0 ∪H. We make a suitable coordinate change in t and
obtain b
(0)
2 , b
(0)
3 , b
(0)
4 as in (6.22). The relations (6.5) and (6.6) still hold. Equation (6.6) implies(
b
(0)
2 b
(0)
4 ̸= 0, β2 + β4 = k1 + k2
)
or
(
b
(0)
3 ̸= 0, 2β3 = k1 + k2
)
(or both). In both cases (6.21) gives
γ =
2
k1 + k2
.
Thus
(β2, β3, β4) =
{(
k1,
k1+k2
2 , k2
)
if k2 > k1,
(k1(1 + 2α3), k1, k1(1− 2α3) if k2 = k1.
(6.23)
In the cases k2 > k1, we have β2 < β3 < β4. Then (6.22) and the relation (6.5) imply b
(0)
2 ̸= 0,
so b
(0)
2,β2
̸= 0. The nonvanishing δ(0) ̸= 0 implies b
(0)
2 b
(0)
4 +
(
b
(0)
3
)2 ̸= 0.
In the case k2−k1 > 0 even, a linear coordinate change in t and a diagonal base change allow
to reduce the triple
(
b
(0)
2,β2
, b
(0)
3,β3
, b
(0)
4,β4
)
∈ C3 to a triple
(
1, ζ, 1− ζ2
)
with ζ ∈ C unique.
In the case k2 − k1 > 0 odd, we have β3 /∈ N, so b(0)3 = 0, and a linear coordinate change in t
and a diagonal base change allow to reduce the pair
(
b
(0)
2,β2
, b
(0)
4,β4
)
∈ (C∗)2 to the pair (1, 1).
In the cases k2 = k1 and α3 ̸= 0, (6.5) and (6.23) imply b
(0)
2 = b
(0)
4 = 0. Then a linear
coordinate change in t allows to reduce b
(0)
3,β3
to the value 1.
In the cases k2 = k1 and α3 = 0, as in the proof of Theorem 6.2(a) (iii), a base change with
constant coefficients leads to b
(0)
2 = b
(0)
4 = 0. Then a linear coordinate change in t allows to
46 C. Hertling
reduce b
(0)
3,β3
to the value 1. In all cases of generic type (Sem), we obtain the normal forms in
the list in Theorem 6.3.
Generic type (Bra). By Theorem 6.2, we can choose the basis v such that b
(1)
2 = b
(1)
4 = 0,
and b
(1)
3 is up to the sign unique. We can choose it to be
b
(1)
3 =
−k2
2(k1 + k2)
∈
(
−1
2
, 0
)
∩Q
(possibly by exchanging v1 and v2). We make a suitable coordinate change in t and obtain b
(0)
2 ,
b
(0)
3 , b
(0)
4 as in (6.22). The nonvanishing δ(1) ̸= 0 and degt δ
(1) = k1 + k2 say
0 ̸= δ(1) = −2b
(1)
3 b
(0)
3 ,
so b
(0)
3 ̸= 0 and
1
γ
= β3 = deg b
(0)
3 = deg δ(1) = k1 + k2, γ =
1
k1 + k2
,
(β2, β3, β4) = (k1, k1 + k2, k1 + 2k2).
The relation (6.5) still holds, and it implies b
(0)
2 ̸= 0. The vanishing δ(0) = 0 says b
(0)
2,β2
b
(0)
4,β4
+(
b
(0)
3,β3
)2
= 0. Together with b
(0)
2,β2
̸= 0 and b
(0)
3,β3
̸= 0 it implies b
(0)
4,β4
̸= 0. A linear coordinate
change in t and a diagonal base change allow to reduce the triple
(
b
(0)
2,β2
, b
(0)
3,β3
, b
(0)
4,β4
)
∈ (C∗)3 to
the triple (1, 1,−1). We obtain the normal form in the list in Theorem 6.3.
Generic type (Reg). First we consider the case when the residue endomorphism of the logarithmic
pole at z = 0 of the (TE)-structure over t0 is semisimple. Then a linear base change gives
b
(1)
2 = b
(1)
4 = 0. And a suitable coordinate change in t gives b
(0)
2 , b
(0)
3 , b
(0)
4 as in (6.22).
First consider the case b
(1)
3 ̸= 0. Then the vanishing 0 = δ(1) = −2b
(1)
3 b
(0)
3 says b
(0)
3 = 0. Now
the vanishing 0 = δ(0) = −b(0)2 b
(0)
4 says that either b
(0)
2 = 0 or b
(0)
4 = 0. Both together cannot be
0 as the generic type is (Reg) and not (Log). After possibly exchanging v1 and v2, we suppose
b
(0)
2 ̸= 0, b
(0)
4 = 0. Now k1 = β2. Write α4 := 2b
(1)
3 ∈ C. By (6.21),
γ =
1 + α4
k1
. (6.24)
A diagonal base change allows to reduce b
(0)
2,β2
to 1.
Now consider the case b
(1)
3 = 0. Then β2 = β3 = β4 = 1/γ, and this is equal to k1, as βj ∈ N
for at least one j. Write α4 := b
(1)
3 = 0. Then (6.24) still holds. By a base change with constant
coefficients, we can obtain b
(0)
2 = tk1 and b
(0)
3 = 0. The vanishing 0 = δ(0) = −b(0)2 b
(0)
4 tells
b
(0)
4 = 0. For α4 ̸= 0 as well as for α4 = 0, we obtain the normal form in the 5th case in the list
in Theorem 6.3.
Finally consider the case when the residue endomorphism of the logarithmic pole at z = 0 of
the (TE)-structure over t0 has a 2 × 2 Jordan block. A base change with constant coefficients
leads to b
(1)
3 = b
(1)
4 = 0 and b
(1)
2 = 1. We will lead the assumption b
(0)
4 ̸= 0 as well as the
assumption b
(0)
4 = 0, b
(0)
3 ̸= 0 to a contradiction.
Assume b
(0)
4 ̸= 0. Denote β4 := degt b
(0)
4 ∈ N. A coordinate change in t leads to b
(0)
4 = −1
β4
tβ4 .
The differential equation in (6.19) for b
(0)
4 gives a
(0)
4 = tβ4−1. Now (6.20) gives b
(0)
3 = −1
β4
ta
(0)
3 .
The differential equation in (6.19) for b
(0)
3 becomes
∂t
(
ta
(0)
3
)
= β4a
(0)
3 + β4t
β4−1.
This equation has no solution in C{t}, a contradiction.
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 47
Assume b
(0)
4 = 0, b
(0)
3 ̸= 0. The same arguments as for the case b
(0)
4 ̸= 0 give a contradiction
if we replace
(
b
(0)
4 , a
(0)
4 , b
(0)
3 , a
(0)
3
)
by
(
b
(0)
3 , a
(0)
3 , b
(0)
2 , a
(0)
2
)
.
Therefore b
(0)
4 = 0, b
(0)
3 = 0, b
(0)
2 ̸= 0. Now k1 = degt b
(0)
2 . A coordinate change in t leads to
b
(0)
2 = tk1 . The differential equations (6.19) gives a
(0)
4 = a
(0)
3 = 0, a
(0)
2 = −k1tk1−1. We obtain
the normal form in the 6th case in the list in Theorem 6.3.
Generic type (Log). Now b
(0)
2 = b
(0)
3 = b
(0)
4 = 0. The cases when all a
(0)
i = 0, were consi-
dered above. We assume now a
(0)
j ̸= 0 for some j ∈ {2, 3, 4}. The equations (6.19) become
a homogeneous system of linear equations with a nontrivial solution. Therefore the determinant
of the 3× 3-matrix in (6.19) vanishes. It is 1− 4
(
b
(1)
3
)2− 4b
(1)
2 b
(1)
4 . Its vanishing tells det
(
B(1)−
zρ(1)C1
)
= −1
4 . As tr
(
B(1) − zρ(1)C1
)
= 0, this matrix has the eigenvalues ±1
2 . Therefore
a linear base change gives
b
(1)
2 = b
(1)
4 = 0, b
(1)
3 = −1
2
.
Now the system of equations (6.19) gives a
(0)
3 = a
(0)
4 = 0, whereas a
(0)
2 is arbitrary in C{t} \ {0}.
Denote k1 := 1+ degt a
(0)
2 ∈ N. A coordinate change in t leads to a
(0)
2 = k1t
k1−1. We obtain the
normal form in the seventh case in the list in Theorem 6.3. ■
6.3 Generically regular singular (TE)-structures over (C, 0)
with logarithmic restriction over t0 = 0 and not semisimple monodromy
The only 1-parameter unfoldings with trace free pole part of logarithmic (TE)-structures over
a point, which are not covered by Theorem 6.3, have generic type (Reg) or (Log) and not
semisimple monodromy. This follows from Theorem 6.2 and Theorem 3.20(a). These (TE)-
structures are classified in Theorem 6.7. Some of them are in the 6th or 9th case in Theorem 6.3,
but most are not.
Theorem 6.7. Consider a rank 2 (TE)-structure
(
H → C×
(
M, t0
)
,∇
)
over
(
M, t0
)
= (C, 0)
which is generically regular singular (so of generic type (Reg) or (Log)), which has trace free pole
part, whose restriction over t0 is logarithmic, and whose monodromy has a 2 × 2 Jordan block.
Associate to it the data in Definition 3.18: H ′ := H|C×(M,t0), M
mon, Nmon, Eig(Mmon) = {λ},
H∞, Cα for α ∈ C with e−2πiα = λ.
The leading exponents of the (TE)-structures over t ̸= 0 come from Theorem 4.5(b) if the
generic type is (Reg) and from Theorem 4.20(b) if the generic type is (Log). In both cases the
leading exponents are independent of t and are still called α1 and α2. Recall α1 − α2 ∈ N0. The
leading exponents of the logarithmic (TE)-structure over t0 = 0 from Theorem 4.20(b) are now
called α0
1 and α0
2. Recall α0
1 − α0
2 ∈ N0.
Precisely one of the three cases (I), (II) and (III) in the following table holds:
case (I) α0
1 = α1 α0
2 = α2 + 1 thus α1 − α2 ∈ N
case (II) α0
1 = α1 + 1 α0
2 = α2
case (III) α0
1 = α1 α0
2 = α2
Choose any section s1 ∈ Cα1 \ ker(∇z∂z − α1 : C
α1 → Cα1). It determines uniquely a section
s2 ∈ ker(∇z∂z − α2 : C
α2 → Cα2) \ {0} with
(∇z∂z − α1)(s1) = zα1−α2s2. (6.25)
48 C. Hertling
Then
O(H)(0,0) = C{t, z}(s1 + fs2)⊕ C{t, z}zs2 for some f ∈ tC{t} \ {0} in case (I), (6.26)
O(H)(0,0) = C{t, z}(s2 + fs1)⊕ C{t, z}zs1 for some f ∈ tC{t} \ {0} in case (II), (6.27)
O(H)(0,0) = C{t, z}s1 ⊕ C{t, z}s2 in case (III). (6.28)
The function f in the cases (6.26) and (6.27) is independent of the choice of s1, so it is an
invariant of the gauge equivalence class of the (TE)-structure.
Before the proof, some remarks are made.
Remarks 6.8.
(i) Equation (6.25) gives
∇z∂z((s1, s2)) = (s1, s2)
(
α1 0
zα1−α2 α2
)
= (s1, s2) · z−1B, (6.29)
with
B = z
α1 + α2
2
C1 + zα1−α2+1C2 + z
α1 − α2
2
D.
(ii) The generic type is (Log) in the case (III). This (TE)-structure is induced by its restriction
over t0 = 0 via the projection φ :
(
M, t0
)
→
{
t0
}
. The matrices A and B for the basis
v = (s1, s2) are A = 0 and B as in (6.29).
(iii) The generic type is (Reg) in the cases (I) and (II). In these cases the (TE)-structure
is induced by the special cases of (6.26) respectively (6.27) with f̃ = t via the map
φ = f : (C, 0) → (C, 0).
(iv) The matrices A and B for the basis v = (s1+fs2, zs2) in (6.26) (⇒ case (I),⇒ α1−α2 ∈ N)
are
A = ∂tf · C2, B =
(
zα1 0
(α2 − α1)f + zα1−α2 z(α2 + 1)
)
. (6.30)
The matrices A and B for the basis v = (s2 + fs1, zs1) in (6.27) (⇒ case (II), ⇒ α1 − α2
∈ N0) are
A = ∂tf · C2, B =
(
zα2 + zα1−α2+1f zα1−α2+2
(α1 − α2)f − zα1−α2f2 z(α1 + 1)− zα1−α2+1f
)
. (6.31)
(v) The invariant k1 ∈ N from (6.1) is here k1 = degt f ∈ N in the case (6.26) and the case
((6.27) and α1 − α2 ∈ N). It is k1 = 2degt f ∈ 2N in the case ((6.27) and α1 = α2).
A suitable coordinate change in t reduces f to f = tk1 respectively f = tk1/2.
(vi) The overlap of the (TE)-structures in Theorem 6.3 and in Theorem 6.7 is as follows.
The case (6.26) with α1 = α2 + 1 and f = −tk1 is the 6th case in Theorem 6.3 with
ρ(1) = α1.
The case (6.28) with α1 = α2 is the 9th case in Theorem 6.3 with ρ(1) = α1.
Proof of Theorem 6.7. Choose any section s01 ∈ Cα0
1 \ ker
(
∇z∂z − α0
1 : C
α0
1 → Cα0
1
)
\ {0}.
It determines uniquely a section s02 ∈ ker
(
∇z∂z : C
α0
2 → Cα0
2
)
\ {0} with(
∇z∂z − α0
1
)
s01 = zα
0
1−α0
2s02.
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 49
Then
O
(
H|C×{t0}
)
0
= C{z}s01|C×{t0} ⊕ C{z}s02|C×{t0}.
Choose a C{t, z}-basis v = (v1, v2) of O(H)(0,0) which extends this C{z}-basis of
O
(
H|C×{t0}
)
0
. It has the shape
v =
(
s01, s
0
2
)
· F with F =
(
f1 f2
f3 f4
)
and
f1, f4 ∈ C{t, z}
[
z−1
]
, f1(z, 0) = f4(z, 0) = 1, f2, f3 ∈ tC{t, z}
[
z−1
]
. (6.32)
We write fj =
∑
k≥degz fj
f
(k)
j zk with f
(k)
j =
∑
l≥0 f
(k)
j,l t
l ∈ C{t} and f
(k)
j,l ∈ C. Also we write
detF =
∑
k≥degz detF
(detF )(k)zk with (detF )(k) ∈ C{t}.
A meromorphic function g ∈ C{t, z}
[
z−1
]
on a neighborhood U ⊂ C×M which is holomorphic
and not vanishing on (U \{0}×M)∪{(0, 0)} is in C{t, z}∗. This and the facts that v and
(
s01, s
0
2
)
are bases of H|U\{0}×M for some neighborhood U ⊂ C×M of (0, 0) and that v|C×{t0} =
(
s01, s
0
2
)
imply
(detF ) ∈ C{t, z}∗, so especially (detF )(k) = 0 for k < 0.
Write kj := degz fj ∈ Z ∪ {∞} (∞ if fj = 0). Recall (6.32). It implies f
(0)
1 , f
(0)
4 ∈ C{t}∗ and
f
(k)
1 , f
(k)
4 ∈ tC{t} for k ̸= 0 and f
(k)
2 , f
(k)
3 ∈ tC{t} for all k. Especially k1 ≤ 0 and k4 ≤ 0.
We distinguish four cases. Precisely one of them holds:
Case (̃I) : 0 = k1 ≤ k2, 0 > min(k3, k4),
Case (̃II) : 0 = k4 ≤ k3, 0 > min(k1, k2),
Case (̃III) : 0 = k1 = k4, 0 ≤ k2, 0 ≤ k3,
Case (̃IV): 0 > min(k1, k2), 0 > min(k3, k4).
We will show: Case (̃I) leads to (6.26) and case (I), case (̃II) leads to (6.27) and case (II),
case (̃III) leads to (6.28) and case (III), and case (̃IV) is impossible.
Case (̃III): Then F ∈ GL2(C{t, z}) and a base change leads to the new basis ṽ =
(
s01, s
0
2
)
. With
(α1, α2, s1, s2) =
(
α0
1, α
0
2, s
0
1, s
0
2
)
,
this gives (6.28) and case (III).
Case (̃I): Then f1 ∈ C{t, z}∗, and a base change leads to a new basis v[1] =
(
s01, s
0
2
)
· F [1] with
f
[1]
1 = 1, f
[1]
2 = 0, f
[1]
4 = detF [1] ∈ C{t, z}∗, f
[1]
3 ∈ tC{t, z}
[
z−1
]
.
As k
[1]
4 = 0, we have k
[1]
3 < 0. A base change leads to a new basis v[2] =
(
s01, s
0
2
)
· F [2] with
F [2] = C1 + f
[2]
3 C2, with f
[2]
3 ∈ tz−1C{t}
[
z−1
]
\ {0}.
The covariant derivative z∇∂tv
[2]
1 = z∂tf
[2]
3 · v[2]2 must be in O(H)0. This shows f
[2]
3 ∈ tz−1C{t}
\ {0}. With
(α1, α2, s1, s2, f) =
(
α0
1, α
0
2 − 1, s01, z
−1s02, zf
[2]
3
)
,
this gives (6.26) and case (II).
50 C. Hertling
Case (̃II): Then f4 ∈ C{t, z}∗, and a base change leads to a new basis v[1] =
(
s01, s
0
2
)
· F [1] with
f
[1]
4 = 1, f
[1]
3 = 0, f
[1]
1 = detF [1] ∈ C{t, z}∗, f
[1]
2 ∈ tC{t, z}
[
z−1
]
.
As k
[1]
1 = 0, we have k
[1]
2 < 0. A base change leads to a new basis v[2] =
(
s01, s
0
2
)
· F [2] with
F [2] = C1 + f
[2]
2 E, with f
[2]
2 ∈ tz−1C{t}
[
z−1
]
\ {0}.
The covariant derivative z∇∂tv
[2]
2 = z∂tf
[2]
2 · v[2]1 must be in O(H)0. This shows f
[2]
2 ∈ tz−1C{t}
\ {0}. With
(α1, α2, s1, s2) =
(
α0
1 − 1, α0
2, z
−1s01, s
0
2
)
,
this gives (6.27) and almost case (II). ”Almost” because we still have to show α1 − α2 ∈ N0.
This follows from the summand −zα1−α2f2 in the left lower entry in the matrix B in (6.31).
Case (̃IV ): Exchange v1 and v2 if k1 > k2 or if k1 = k2 and degt f
(k1)
1 > degt f
(k1)
2 . Keep the
basis v if not. The new basis v[1] satisfies min(k1, k2) = k
[1]
1 ≤ k
[1]
2 , and in the case k
[1]
1 = k
[1]
2 it
satisfies degt
(
f
[1]
1
)(k[1]1 ) ≤ degt
(
f
[1]
2
)(k[1]1 )
.
By replacing v
[1]
2 by a suitable element in v
[1]
2 +C{t, z}v[1]1 , we obtain a new basis v[2] either
with f
[2]
2 = 0 or with k
[2]
1 < k
[2]
2 and degt
(
f
[2]
1
)(k[2]1 )
> degt
(
f
[2]
2
)(k[2]2 )
.
The case f
[2]
2 = 0 is impossible, as then we would have f
[2]
1 f
[2]
4 = detF [2] ∈ C{t, z}∗, so
f
[2]
1 ∈ C{t, z}∗ and 0 = k
[2]
1 , but also k
[2]
1 = k
[1]
1 = min(k1, k2) < 0. For the same reason, f
[2]
3 = 0
is impossible.
f
[2]
4 = 0 is impossible as then we would have −f [2]2 f
[2]
3 = detF [2] ∈ C{t, z}∗, so f [2]2 , f
[2]
3 ∈
C{t, z}∗, 0 = k
[2]
2 = k
[2]
3 and k
[2]
4 = ∞, so 0 = min
(
k
[2]
3 , k
[2]
4
)
= min(k3, k4) < 0, a contradiction.
Write
l2 := degt
(
f
[2]
2
)(k[2]2 ) ∈ N0, l1 := degt
(
f
[2]
1
)(k[2]1 ) − l2 ∈ N,
l3 := degt
(
f
[2]
3
)(k[2]3 ) ∈ N0, l4 := degt
(
f
[2]
4
)(k[2]4 ) ∈ N0.
Multiplying v
[2]
1 and v
[2]
2 by suitable units in C{t}, we obtain a basis v[3] with k
[3]
j = k
[2]
j and
(
f
[3]
1
)(k[3]1 )
= tl1+l2 ,
(
f
[3]
2
)(k[3]2 )
= tl2 ,(
f
[3]
3
)(k[3]3 )
= tl3 · u3,
(
f
[3]
4
)(k[3]4 )
= tl4 · u4
for some units u3, u4 ∈ C{t}∗. We still have 0 > k
[3]
1 < k
[3]
2 and min
(
k
[3]
3 , k
[3]
4
)
< 0. Consider
z∇∂t
(
v
[3]
1
)
= z∂tf
[3]
1 · s01 + z∂tf
[3]
3 · s02 ∈ O(H)(0,0) = C{t, z}v[3]1 ⊕ C{t, z}v[3]2 .
The leading nonvanishing monomial in z∂tf
[3]
1 is zk
[3]
1 +1tl1+l2−1. This implies k
[3]
2 = k
[3]
1 +1 ≤ 0.
Therefore k
[3]
1 +k
[3]
4 < 0 or k
[3]
2 +k
[3]
3 < 0. Each part
(
detF [3]
)(k)
for k < 0 vanishes. This shows
k
[3]
1 + k
[3]
4 = k
[3]
2 + k
[3]
3 < 0, so k
[3]
4 = k
[3]
3 + 1 ≤ 0, k
[3]
3 < 0,
0 =
(
f
[3]
1
)(k[3]1 )(
f
[3]
4
)(k[3]3 +1) −
(
f
[3]
2
)(k[3]1 +1)(
f
[3]
3
)(k[3]3 )
= tl2
(
tl1+l4u4 − tl3u3
)
,
so l3 = l1 + l4, u3 = u4.
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 51
We can write
v[3] =
(
s01, s
0
2
)( (tl1+l2 + zg1)z
k
[3]
1 (tl2 + zg2)z
k
[3]
1 +1
(tl1+l4u3 + zg3)z
k
[3]
3 (tl4u3 + zg4)z
k
[3]
3 +1
)
with some suitable g1, g2, g3, g4 ∈ C{t, z}. This shows
O(H)(0,0) ∩
(
zk
[3]
1 +2C{t, z}s01 + C{t, z}
[
z−1
]
s02
)
= z2C{t, z}v[3]1 + zC{t, z}v[3]2 + C{t, z}
(
zv
[3]
1 − tl1v
[3]
2
)
⊂ O(H)(0,0) ∩
(
C{t, z}
[
z−1
]
s01 + zk
[3]
3 +2C{t, z}s02
)
. (6.33)
Now consider the element
z
(
∇z∂z −
(
α0
1 + k
[3]
1
))(
v
[3]
1
)
= z2∂z(zg1)z
k
[3]
1 s01 +
(
tl1+l2 + zg1
)
zk
[3]
1 +1+α0
1−α0
2s02 + z2∂z(zg3)z
k
[3]
3 s02
+
(
tl1+l4u3 + zg3
)(
k
[3]
3 + α0
2 − α0
1 − k
[3]
1
)
zk
[3]
3 +1s02
of O(H)(0,0). It is contained in the first line of (6.33), and therefore also in the third line
of (6.33). But this leads to a contradiction, when we compare the coefficient of s02. Here observe
k
[3]
3 + α0
2 − α0
1 − k
[3]
1
<
=
>
0 ⇐⇒ k
[3]
1 + 1 + α0
1 − α0
2
<
=
>
k
[3]
3 + 1.
This contradiction shows that case (̃IV) is impossible. ■
7 Marked regular singular rank 2 (TE)-structures
The regular singular rank 2 (TE)-structures over points were subject of Sections 4.5 and 4.6,
those over (C, 0) were subject of Theorem 6.3 and Remark 6.4(iv) and of Theorem 6.7 and Re-
marks 6.8.
First we will consider in Remarks 7.1(i)+(ii) regular singular rank 2 (TE)-structures over P1,
which arise naturally from Theorems 4.17 and 4.20. The (TE)-structures over the germs
(
P1, 0
)
and
(
P1,∞
)
appeared already in Remark 6.4(iv) and in Theorem 6.7.
With the construction in Lemma 3.10(d), each of these (TE)-structures over P1 extends to
a rank 2 (TE)-structure of generic type (Reg) or (Log) over C× P1 with primitive Higgs field.
With Theorem 3.14, the base manifold C×P1 obtains a canonical structure as F -manifold with
Euler field. For each t0 ∈ C× C∗, the (TE)-structure over the germ
(
C× P1, t0
)
is a universal
unfolding of its restriction over t0. For each t0 ∈ C× {0,∞}, the (TE)-structure over the germ(
C× P1, t0
)
will reappear in Theorems 8.1, 8.5 and 8.6. See Remarks 7.2(i)+(ii).
Then we will observe in Corollary 7.3 that any marked regular singular (TE)-structure is
a good family of marked regular singular (TE)-structures (over points) in the sense of Defini-
tion 3.26(b).
In Theorem 7.4 we will determine the moduli spaces M (Href,∞,Mref),reg for marked regular
singular rank 2 (TE)-structures, which were subject of Theorem 3.29. The parameter space P1
of each (TE)-structure over P1 in Remarks 7.1(i)+(ii) embeds into one of these moduli spaces,
after the choice of a marking. Also these embeddings will be described in Theorem 7.4.
Because of Corollary 7.3, any marked regular singular rank 2 (TE)-structure over a mani-
fold M is induced by a holomorphic map M → M (Href,∞,Mref),reg, where
(
Href,∞,M ref
)
is the
reference pair used in the marking of the (TE)-structure. Remark 7.5 says something about the
horizontal direction(s) in the moduli spaces.
52 C. Hertling
Remarks 7.1.
(i) Consider the manifold M (3) := P1 with coordinate t2 on C ⊂ P1 and coordinate t̃2 := t−1
2
on P1 \ {0} ⊂ P1.
With the projection M (3) → {0}, we pull back the flat bundle H ′ → C∗ in Theorem 4.17
to a flat bundle H(3)′ on C∗ ×M . Recall the notations (4.4).
Now we read v := (s1 + t2s2, zs2) in (4.25) and (4.27) as a basis of sections on H(3)′|C∗×C,
and ṽ :=
(
s2 + t̃2s1, zs1
)
in (4.26) and (4.19) as a basis of sections on H(3)′|C∗×(P1\{0}).
One sees immediately
z∇∂2v = v C2, z∇
∂̃2
ṽ = ṽ C2. (7.1)
and again (4.25) resp. (4.27) and (4.26) resp. (4.19). Therefore v and ṽ are in any case
bases of a (TE)-structure
(
H(3) → C ×M (3),∇(3)
)
on C × C ⊂ C ×M (3) respectively
C×
(
P1 \{0}
)
⊂ C×M (3). The restricted (TE)-structures over t2 ∈ C∗ are those in Theo-
rem 4.17. They are regular singular, but not logarithmic. Their leadings exponents α1
and α2 are independent of t2 ∈ C∗. The (TE)-structures over t2 = 0 and over t̃2 = 0
(so t2 = ∞) are logarithmic except for the case (Nmon ̸= 0 and α1 = α2), in which case
the one over t2 = 0 is regular singular, but not logarithmic. Their leading exponents are
called α0
1 and α0
2 and α∞
1 and α∞
2 . Then
over 0 over ∞
α0
1 = α1 α∞
1 = α1 + 1
α0
2 = α2 + 1 α∞
2 = α2
except that in the case (Nmon ̸= 0 and α1 = α2) we have α0
1 = α1, α
0
2 = α2. For use
in Theorem 7.3, we write the base space for the (TE)-structure over P1 with leading
exponents α1 and α2 as M (3),0,α1,α2 ∼= P1 in the case Nmon = 0 and as M (3),̸=0,α1,α2 ∼= P1
in the case Nmon ̸= 0.
(ii) We extend the case Nmon = 0 from Theorem 4.17(a) to the case α1 = α2. (4.25) and (4.26)
still hold, but now the restricted (TE)-structures over points inM (3) = P1 are all logarith-
mic, though the (TE)-structure over M (3) is not logarithmic, but only regular singular.
(7.1) still holds. In this case, the leading exponents are constant and are α1 and α1 + 1
(so, not α1 and α2 = α1). Similarly to (i), the base space is called M (3),0,α1,log ∼= P1.
Remarks 7.2.
(i) The construction in Lemma 3.10(d) extends a (TE)-structure
(
H(3) → C ×M (3),∇(3)
)
in Remark 7.1(i) or (ii) with M (3) = P1 to a (TE)-structure
(
H(4) → C ×M (4),∇(4)
)
with M (4) = C ×M (3) = C × P1, via
(
O
(
H(4)
)
,∇(4)
)
=
(
φ(4)
)∗(O(H(3)
)
,∇(3)
)
⊗ E t1/z,
where t1 is the coordinate on the first factor C in C× P1, and where φ(4) : M (4) → M (3),
(t1, t2) 7→ t2, is the projection. Define v(4) :=
(
φ(4)
)∗
(v in Remark 7.1(i) or (ii)).
Then the matrices A2 and B with z∇∂iv
(4) = v(4)Ai and z
2∇∂zv
(4) = v(4)B are unchanged,
and A1 = C1, so
A1 = C1, A2 = C2 (as in (7.1)), B is as in (4.25) or (4.27).
The Higgs field is everywhere onM (4) primitive. By Theorem 3.14,M (4) = C×P1 is an F -
manifold with Euler field. The unit field is ∂1, the multiplication is given by ∂2 ◦ ∂2 = 0
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 53
and ∂̃2 ◦ ∂̃2 = 0. So, each germ
(
M (4), t0
)
is the germ N2. The Euler field is
E = t1∂1 + (α1 − α2)t2∂2 = t1∂1 + (α2 − α1)t̃2∂̃2{
in the case (4.25) and (4.26) and in (ii) above,
and in the case (4.27) and (4.19) with α1 − α2 ∈ N,
E = t1∂1 − ∂2 = t1∂1 + t̃22∂̃2 in the case (4.27) and (4.19) with α1 = α2.
(ii) For t(4) ∈ C× C∗ ⊂ M (4), the (TE)-structure
(
H(4) → C×
(
M (4), t(4)
)
,∇
)
is a universal
unfolding of the one over t0, because that one is of type (Reg) and because the Higgs field
is primitive. See Corollary 5.1.
(iii) Let
(
H → C ×
(
M, t0
)
,∇
)
be a regular singular unfolding of a regular singular, but not
logarithmic rank 2 (TE)-structure over t0. Because of part (ii), it is induced by the (TE)-
structure
(
H(4),∇(4)
)
via a map
(
M, t0
)
→
(
M (4), t(4)
)
for some t(4) ∈ {0} ×C∗. Because
it is regular singular, the image of the map is in {0} × C∗ ⊂ {0} ×M (3). As there the
leading exponents are constant, they are also constant on the unfolding (H,∇).
Theorem 7.3. Any marked regular singular rank 2 (TE)-structure (see Definition 3.15(b), espe-
cially, M is simply connected) is a good family of marked regular singular (TE)-structures (over
points) in the sense of Definition 3.26(b).
Proof. Let ((H → C×M,∇), ψ) be a regular singular rank 2 (TE)-structure with a marking ψ,
i.e., an isomorphism ψ from (H∞,Mmon) to a reference pair
(
Href,∞,M ref
)
. We have to show
the conditions (3.21) and (3.22) for a good family of marked regular singular (TE)-structures.
By definition of a marking, M is simply connected, so especially it is connected. The subset
M [log] := {t ∈M | U|t = 0} = {t ∈M | the (TE)-structure over t is logarithmic}
is a priori a subvariety (in fact, it is either ∅ or a hypersurface or equal to M).
First consider the caseM [log] =M . Choose any point t0 ∈M and any disk ∆ ⊂M through t0.
The restriction of the (TE)-structure over the germ
(
∆, t0
)
is in the case Nmon = 0 isomorphic
to one in the 7th or 8th or 9th case in Theorem 6.3. In the case Nmon ̸= 0, it is isomorphic to one
in case (III) in Theorem 6.7. In either case the leading exponents are constant on ∆, because
of table (6.13) in Remark 6.4(iv) and because of the definition of case (III) in Theorem 6.7.
Therefore they are constant on M . We call them αgen
1 and αgen
2 .
Now consider the case M [log] ⫋ M . For each t0 ∈ M \M [log], the (TE)-structure over the
germ
(
M, t0
)
has constant leading exponents because of Remark 7.2(iii). Therefore the leading
exponents are constant on M \M [log]. We call these generic leading exponents αgen
1 and αgen
2 .
For t0 ∈M [log] choose a generic small disk ∆ ⊂M through t0. Then ∆ \
{
t0
}
⊂M \M [log].
The restriction of the (TE)-structure over the germ
(
∆, t0
)
is in the case Nmon = 0 isomorphic
to one in the 5th case in Theorem 6.3. In the case Nmon ̸= 0, it is isomorphic to one in
case (I) or case (II) in Theorem 6.7. In either case, the leading exponents
(
α1
(
t0
)
, α2
(
t0
))
of
the (TE)-structure over t0 are either
(
αgen
1 +1, αgen
2
)
or
(
αgen
1 , αgen
2 +1
)
, because of table (6.13)
in Remark 6.4(iv) and because of the definition of the cases (I) and (II) in Theorem 6.7.
Remark 6.4(iv) and Theorem 6.7 provide generators of O
(
H|C×(∆,t0)
)
(0,t0)
which are cer-
tain linear combinations of elementary sections. The shape of these generators and the almost
constancy of the leading exponents imply the two conditions,
O
(
H|C×{t}
)
(0,t)
⊃ V r for any t ∈M, where r := max
(
Re
(
αgen
1
)
+ 1, Re
(
αgen
2
)
+ 1
)
,
dimCO
(
H|C×{t}
)
(0,t)
/V r is independent of t ∈M,
which are the conditions (3.21) and (3.22) for a good family of marked regular singular (TE)-
structures. ■
54 C. Hertling
The following theorem describes the moduli space M (Href,∞,Mref),reg from Theorem 3.29 for
the marked regular singular rank 2 (TE)-structures as infinite unions of curves isomorphic
to P1 such that the families of (TE)-structures over these curves are the (TE)-structures in
Remarks 7.1(i)+(ii). Part (a) treats the cases with Nmon = 0, part (b) treats the cases with
Nmon ̸= 0. Recall the definitions ofM (3),0,α1,α2 , M (3),̸=0,α1,α2 andM (3),0,α1,log in Remarks 7.1(i)
and (ii).
Theorem 7.4. Let
(
Href,∞,M ref
)
be a reference pair with dimHref,∞ = 2. Let Eig
(
M ref
)
=
{λ1, λ2} be the set of eigenvalues ofM ref . Let β1, β2 ∈ C be the unique numbers with e−2πiβj = λj
and −1 < Reβj ≤ 0.
(a) The case Nmon = 0.
(i) The cases with λ1 ̸= λ2. Then
M (Href,∞,Mref),reg =
⋃̇
l1∈Z
( ⋃
l2∈Z
M (3),0,β1+l1+l2,β2−l2
)
.
Its topological components are the unions in brackets, so
⋃
l2∈ZM
(3),0,β1+l1+l2,β2−l2.
Each component is a chain of P1’s, the point ∞ of M (3),0,α1,α2 is identified with the
point 0 of M (3),0,α1+1,α2−1.
u u u u
(
α1 − 1
α2 + 2
)(
α1 − 1
α2 + 1
)( α1
α2 + 1
) (
α1
α2
) (
α1 + 1
α2
)(
α1 + 1
α2 − 1
)(α1 + 2
α2 − 1
)
(Log) (Log) (Log) (Log)
(Reg) (Reg) (Reg)6 6 6 6
? ? ? ?
Figure 7.1. One topological component in part (a) (i).
(ii) The cases with λ1 = λ2 (so β1 = β2). Then
M (Href,∞,Mref),reg =
⋃̇
l1∈Z
( ⋃
l2∈N
Fβ1+l1+l2,β1+l1−l2
2
)
∪
⋃̇
l1∈Z
(
F̃β1+l1+1,β1+l1
2 ∪
⋃
l2∈N
Fβ1+l1+1+l2,β1+l1−l2
2
)
. (7.2)
Here Fα1,α2
2 is for all possible α1, α2 the Hirzebruch surface F2, and F̃α1,α1−1
2 is the
surface F̃2, which is obtained from F2 by blowing down the unique (−2)-curve in F2.
The unions in brackets are the topological components. They are chains of Hirzebruch
surfaces. A (+2)-curve of Fα1,α2
2 is identified with the (−2)-curve of Fα1+1,α2−1
2 (and
a (+2)-curve of F̃α1,α1−1
2 is identified with the (−2) curve in Fα1+1,α1−2
2 ). The (TE)-
structures over the points in the (−2)-curves are logarithmic, and also the (TE)-
structure over the singular point of F̃α1,α1−1
2 is logarithmic. The (TE)-structures over
all other points of Fα1,α2
2 and F̃α1,α2
2 are regular singular, but not logarithmic, and have
leading exponents α1 and α2. For each Fα1,α2
2 , and also for F̃α1,α2
2 after blowing up
the singular point to a (−2)-curve, the fibers of it as a P1-fiber bundle over P1 are
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 55
isomorphic to M (3),0,α1,α2. The (−2)-curve in Fα1,α1−2
2 (the F2 with l2 = 1 in each
topological component in the first line of (7.2)) is isomorphic to M (3),0,α1−1,log, and
the (TE)-structures over its points are logarithmic with leading exponents α1, α1− 1.
u u u u
u u u u
u u u u
(
α1
α1 − 1
)(
α1
α1 − 2
)(α1 + 1
α1 − 2
)(
α1 + 1
α1 − 3
)(α1 + 2
α1 − 3
)(
α1 + 2
α1 − 4
)(α1 + 3
α1 − 4
)
(Log) (Log) (Log)
(Reg) (Reg) (Reg)
M (3),0,α1−1,log
(Log)
6 6 6 6
? ? ? ?
Fα1,α1−2
2 Fα1+1,α1−3
2 Fα1+2,α1−4
2
Figure 7.2. One topological component in part (a) (ii).
u u u
u u u u
u u u
(
α1
α1
) (
α1
α1 − 1
)(α1 + 1
α1 − 1
)(
α1 + 1
α1 − 2
)(α1 + 2
α1 − 2
)(
α1 + 2
α1 − 3
)(α1 + 3
α1 − 3
)
(Log) (Log) (Log) (Log)
(Reg) (Reg) (Reg)
6 6 6
? ? ?
6
?
F̃α1,α1−1
2 Fα1+1,α1−2
2 Fα1+2,α1−3
2
Figure 7.3. Another topological component in part (a) (ii).
(b) The cases with Nmon ̸= 0 (and thus λ1 = λ2, β1 = β2). Then
M (Href,∞,Mref),reg =
⋃̇
l1∈Z
( ⋃
l2∈N0
M (3),̸=0,β1+l1+l2,β1+l1−l2
)
∪
⋃̇
l1∈Z
( ⋃
l2∈N0
M (3),̸=0,β1+l1+1+l2,β1+l1−l2
)
.
56 C. Hertling
Its topological components are the unions in brackets. Each component is a chain of P1’s,
the point ∞ of M (3), ̸=0,α1,α2 is identified with the point 0 of M (3),̸=0,α1+1,α2−1.
u u u u
(
α1
α1
) (
α1 + 1
α1
)(
α1 + 1
α1 − 1
)(α1 + 2
α1 − 1
)(
α1 + 2
α1 − 2
)(α1 + 3
α1 − 2
)
(Log) (Log) (Log)
(Reg) (Reg) (Reg)6 6 6
? ? ?
Figure 7.4. One topological component in part (b).
u u u u
(
α1
α1
) (
α1
α1 − 1
)(α1 + 1
α1 − 1
)(
α1 + 1
α1 − 2
)(α1 + 2
α1 − 2
)(
α1 + 2
α1 − 3
)(α1 + 3
α1 − 3
)
(Log) (Log) (Log) (Log)
(Reg) (Reg) (Reg)6 6 6 6
? ? ? ?
Figure 7.5. Another topological component in part (b).
Proof. We consider only marked (TE)-structures with a fixed reference pair
(
Href,∞,M ref
)
.
Because of the markings, we can identify for each such (TE)-structure its pair (H∞,Mmon)
with the reference pair
(
Href,∞,M ref
)
. Thus also the spaces Cα can be identified for all marked
(TE)-structures.
(a) (i) and (b) In both parts, there is no harm in fixing elementary sections s1 ∈ Cα1 and
s2 ∈ Cα2 as in Theorem 4.17. Then Theorem 4.17 lists all marked (TE)-structures with the given
reference pair. Remarks 7.1(i) just put these marked (TE)-structures into families parametrized
by the spaces M (3),0,α1,α2 resp. M (3), ̸=0,α1,α2 . Most logarithmic (TE)-structures (which are
classified in Theorem 4.20) turn up in two such families. This leads to the identification of the
point ∞ in M (3),0/̸=0,α1,α2 with the point 0 in M (3),0/ ̸=0,α1+1,α2−1. Only each of the logarithmic
(TE)-structures with Nmon ̸= 0 and leading exponents α1 = α2 turns up in only one P1, in the
space M (3), ̸=0,α1,α1−1. There it is over the point 0.
(a) (ii) Here the leading exponents satisfy α1 − α2 ∈ Z \ {0}, and we index them such that
α1−α2 ∈ N. We fix a basis σ1, σ2 of C
α1 and define σ3 := zα2−α1σ1 ∈ Cα2 , σ4 = zα2−α1σ2 ∈ Cα2 .
Then because of Theorem 4.17(a), we can write all marked regular singular, but not logarithmic
(TE)-structures with leading exponents α1 and α2 in two charts C×C∗ with coordinates (r1, t2)
and (r2, t3),
O(H)0 = C{z}(σ1 + t2(σ4 + r1σ3))⊕ C{z}(z(σ4 + r1σ3)),
O(H)0 = C{z}(σ2 + t3(σ3 + r2σ4))⊕ C{z}(z(σ3 + r2σ4)).
The charts overlap where r1, r2 ∈ C∗, with
r2 = r−1
1 , t3 = −t2r21.
Compactification to t2 = 0 and t2 = ∞ (and t3 = 0 and t3 = ∞) gives the Hirzebruch surface
F2 = Fα1,α2
2 . The curve with t2 = 0 (and t3 = 0) is the (−2)-curve. Over this curve, we have
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 57
the family of marked logarithmic (TE)-structures (see Theorem 4.20) with leading exponents α1
and α2 + 1,
O(H)0 = C{z}(σ1)⊕ C{z}(z(σ4 + r1σ3)),
O(H)0 = C{z}(σ2)⊕ C{z}(z(σ3 + r2σ4)).
The curve with t2 = ∞ (and t3 = ∞) is a (+2)-curve. Over this curve, we have the family
of marked logarithmic (TE)-structures with leading exponents α1 + 1 and α2. Therefore the
(+2)-curve in Fα1,α2
2 must be identified with the (−2)-curve in Fα1+1,α2−1
2 .
In the case α1 − α2 = 2, the (−2)-curve in Fα1,α2
2 is the moduli space M (3),0,α1−1,log from
Remark 7.1(i).
In the case α1 − α2 = 1, the (−2)-curve in Fα1,α2
2 has to be blown down, as then for t2 = 0
O(H)0 = C{z}(σ1)⊕ C{z}(z(σ4 + r1σ3)) = C{z}Cα1 = V α1
is independent of the parameter r1.
The projection (r1, t2) 7→ r1 extends to the P1-fibration of Fα1,α2
2 over P1. The fibers are
isomorphic to M (3),0,α1,α2 . Affine coordinates on these fibers are t2 and t̃2 = t−1
2 or t3 and
t̃3 = t−1
3 . ■
Remarks 7.5.
(i) Consider a marked regular singular rank 2 (TE)-structure ((H → C ×M,∇), ψ). There
is a unique map φ : M → M (Href,∞,Mref),reg, which maps t ∈ M to the unique point
inM (Href,∞,Mref),reg over which one has up to marked isomorphism the same marked (TE)-
structure as over t. Corollary 7.3 and the fact that the moduli space represents the moduli
functor M(Href,∞,Mref),reg, imply that φ is holomorphic.
Because M is (simply) connected, the map φ goes to one irreducible component of the
moduli space, so to one M (3),0/̸=0,α1,α2 ∼= P1 in the parts (a) (i) and (b) in Theorem 7.4
and to one Fα1,α2
2 or to F̃α1,α1−1
2 in part (a) (ii).
(ii) In fact, in part (a) (ii) the map φ goes even to a projective curve which is isomorphic to
one curve M (3),0,α1,α2 or to the curve M (3),0,α1,log. This holds for the (TE)-structure over
any manifold, as it holds by Remark 6.4(iv) and Theorem 6.7 for the (TE)-structures over
the 1-dimensional germ (C, 0).
The curves isomorphic toM (3),0,α1,α2 are the (0)-curves in the P1 fibration of Fα1,α2
2 over P1(
in the case of F̃α2+1,α2
2 each fiber of Fα2+1,α2
2 over P1 embeds also into the blown down
surface F̃α2+1,α2
2
)
. The curve isomorphic to M (3),0,α1,log is the (−2)-curve in Fα1,α1−2
2 .
(iii) We have here a notion of horizontal directions which is similar to that for classifying spaces
of Hodge structures. There it comes from Griffiths transversality. Here it comes from the
part of the pole of Poincaré rank 1, which says that the covariant derivatives ∇∂j along
vector fields on the base space see only a pole of order 1.
In the cases of the Fα1,α2
2 with α1 − α2 ∈ N \ {1, 2}, the horizontal directions are the
tangent spaces to the fibers of the P1 fibration. In the cases of Fα1,α1−2
2 and F̃α1,α1−1
2 , the
horizontal directions contain these tangent spaces. However, on points in the (−2)-curve
in Fα1,α1−2
2 and on the singular point in F̃α1,α1−1
2 , any direction is horizontal.
Remark 7.6. If we forget the markings of the (TE)-structures in one moduli space
M (Href,∞,Mref),reg and consider the unmarked (TE)-structures up to isomorphism, we obtain
in the cases Nmon = 0 countably many points, one for each intersection point or intersection
curve of two irreducible components, and one for each irreducible component. On the contrary,
58 C. Hertling
in the cases Nmon ̸= 0, the unmarked and the marked (TE)-structures almost coincide, as the
choice of an elementary section s1 in Theorem 4.17(b) fixes uniquely the elementary section s2
with (4.22). The set of unmarked (TE)-structures up to isomorphism is still almost in bijec-
tion with the moduli space M (Href,∞,Mref),reg in the case Nmon ̸= 0. Only the components
M (3), ̸=0,α1,α1 − {∞} boil down to single points.
8 Unfoldings of rank 2 (TE)-structures of type (Log)
over a point
Sections 5 and 8 together treat all rank 2 (TE)-structures over germs
(
M, t0
)
of manifolds.
Section 5 treated the unfoldings of (TE)-structures of types (Sem) or (Bra) or (Reg) over t0.
Section 8 will treat the unfoldings of (TE)-structures of type (Log) over t0.
It builds on Section 6, which classified the unfoldings with trace free pole parts over
(
M, t0
)
=
(C, 0) of a logarithmic rank 2 (TE)-structure over t0 and on Section 7, which treated arbitrary
regular singular rank 2 (TE)-structures. Here Lemmata 3.10 and 3.11 are helpful. They allow to
go from arbitrary (TE)-structures to (TE)-structures with trace free pole parts and vice versa.
Section 8.1 gives the classification results. Section 8.2 extracts from them a characterization
of the space of all (TE)-structures with generically primitive Higgs fields over a given germ of
a 2-dimensional F -manifold with Euler field. Section 8.3 gives the proof of Theorem 8.5.
First we characterize in Theorem 8.1 the 2-parameter unfoldings of rank 2 (TE)-structures
of type (Log) over a point such that the Higgs field is generically primitive and induces an
F -manifold structure on the underlying germ
(
M, t0
)
of a manifold. Theorem 8.1 is a rather
immediate implication of Theorem 6.3 and Theorem 6.7 together with Lemmata 3.10 and 3.11.
Part (d) gives an explicit classification. The other results in this section will all refer to this
classification.
Corollary 8.3 lists for any logarithmic rank 2 (TE)-structure over a point t0 all unfoldings
within the set of (TE)-structures in Theorem 8.1(a). The proof consists of inspection of the
explicit classification in Theorem 8.1(d).
Theorem 8.5 is the main result of this section. It lists a finite subset of the unfoldings in Theo-
rem 8.1(d) with the following property: Any unfolding of a rank 2 (TE)-structure of type (Log)
over a point is induced by a (TE)-structure in this list. The (TE)-structures in the list turn out
to be universal unfoldings of themselves.
The proof of Theorem 8.5 is long. It is deferred to Section 8.3. The results of Section 6 are
crucial, especially Theorem 6.3 and Theorem 6.7.
Finally, Theorem 8.6 lists the rank 2 (TE)-structures over a germ
(
M, t0
)
of a manifold such
that the Higgs field is primitive (so that
(
M, t0
)
becomes a germ of an F -manifold with Euler
field) and the restriction over t0 is of type (Log). This list turns out to be a sublist of the one
in Theorem 8.5. Theorem 8.6 follows easily from Theorem 8.1.
Theorem 8.6 is also contained in the papers [6] and [7], the generic types (Bra), (Reg) and
(Log) are in [6], the generic type (Sem) is in [7]. The proofs there are completely different. They
build on the formal classification of (T )-structures in [5].
8.1 Classification results
Theorem 8.1.
(a) Consider a rank 2 (TE)-structure
(
H → C×
(
M, t0
)
,∇
)
over a 2-dimensional germ
(
M, t0
)
with restriction over t0 of type (Log), with generically primitive Higgs field, and such that
the induced F -manifold structure on generic points of M extends to all of M .
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 59
There is a unique rank 2 (TE)-structure
(
H [3] → C × (C, 0),∇[3]
)
over (C, 0) (with co-
ordinate t2) with trace free pole part, with nonvanishing Higgs field and with logarithmic
restriction over t2 = 0 such that (O(H),∇) arises from
(
O(H [3]),∇[3]
)
as follows. There
are coordinates t = (t1, t2) on
(
M, t0
)
such that
(
M, t0
)
=
(
C2, 0
)
and a constant c1 ∈ C
such that
(O(H),∇) ∼= pr∗2
(
O
(
H [3]
)
,∇[3]
)
⊗ E(t1+c1)/z, (8.1)
where pr2 :
(
M, t0
)
→ (C, 0), (t1, t2) 7→ t2
(
see Lemma 3.10(a) for E(t1+c1)/z
)
.
The (TE)-structure (H,∇) is of type (Log) over (C×{0}, 0) and of one generic type (Sem)
or (Bra) or (Reg) or (Log) over (C× C∗, 0).
(b) Vice versa, if
(
H [3],∇[3]
)
is as in (a) and c1 ∈ C, then the (TE)-structure (O(H),∇) :=
pr∗2
(
O
(
H [3]
)
,∇[3]
)
⊗ E(t1+c)/z over
(
M, t0
)
=
(
C2, 0
)
satisfies the properties in (a).
(c) The rank 2 (TE)-structures
(
H [3],∇[3]
)
over (C, 0) with trace free pole part, nonvanishing
Higgs field and logarithmic restriction over 0 are classified in Theorems 6.3 and 6.7. They
are in suitable coordinates the first 7 of the 9 cases in the list in Theorem 6.3 and the
cases (6.26) and (6.27) with f = 1
k1
tk1 for some k1 ∈ N in Theorem 6.7. (Though here the
6th case in Theorem 6.3 is part of the cases (6.26) and (6.27) in Theorem 6.7.)
(d) The explicit classification of the (TE)-structures (H,∇) in (a) is as follows. There are
coordinates (t1, t2) such that
(
M, t0
)
=
(
C2, 0
)
, and there is a C{t, z}-basis v of O(H)0
whose matrices A1, A2, B ∈ M2×2(C{t, z}) with z∇∂iv = vAi, z
2∇∂zv = vB are in the
following list of normal forms. The normal form is unique. We always have
A1 = C1.
Always M is an F -manifold with Euler field in one of the normal forms in Theorems 2.2
and 2.3 (in the case (i) the product ∂2 ◦ ∂2 is only almost in the normal form in Theo-
rem 2.2; in the case (iii) with α4 = −1 the Euler field is only almost in the normal form
in Theorem 2.3).
(i) Generic type (Sem): invariants k1, k2 ∈ N with k2 ≥ k1, c1, ρ
(1) ∈ C, ζ ∈ C if
k2 − k1 ∈ 2N, α3 ∈ R≥0 ∪H if k1 = k2,
γ :=
2
k1 + k2
,
A2 =
−γ−1
(
tk1−1
2 C2 + tk2−1
2 E
)
if k2 − k1 > 0 is odd,
−γ−1
(
tk1−1
2 C2 + ζt(k1+k2)/2−1D +
(
1− ζ2
)
tk2−1
2 E
)
if k2 − k1 ∈ 2N,
−γ−1tk1−1
2 D if k2 = k1,
B =
(
−t1 − c1 + zρ(1)
)
C1 + (−γt2)A2 +
{
z k1−k2
2(k1+k2)
D if k2 > k1,
zα3D if k2 = k1,
F -manifold I2(k1 + k2)
(
with I2(2) = A2
1
)
, with ∂2 ◦ ∂2 = γ−2tk1+k2
2 · ∂1,
E = (t1 + c1)∂1 + γt2∂2 Euler field.
(ii) Generic type (Bra): invariants k1, k2 ∈ N, c1, ρ(1) ∈ C,
γ :=
1
k1 + k2
,
A2 = −γ−1
(
tk1−1
2 C2 + tk1+k2−1
2 D − tk1+2k2−1
2 E
)
,
60 C. Hertling
B =
(
−t1 − c1 + zρ(1)
)
C1 + (−γt2)A2 + z
−k2
2(k1 + k2)
D,
F -manifold N2, with ∂2 ◦ ∂2 = 0,
E = (t1 + c1)∂1 + γt2∂2 Euler field.
(iii) Generic type (Reg): invariants c1, ρ
(1) ∈ C, α4 ∈ C \ {−1} if Nmon = 0, α4 ∈ Z
if Nmon ̸= 0, k1 ∈ N if Nmon = 0, k̃1 ∈ N if Nmon ̸= 0 (with k1 = k̃1 if α4 ̸= −1, and
k1 = 2k̃1 if α4 = −1),
γ :=
1 + α4
k1
,
A2 =
{
−γ−1tk1−1
2 C2 if Nmon = 0,
k̃1t
k̃1−1
2 C2 if Nmon ̸= 0,
B =
(
−t1 − c1 + zρ(1)
)
C1 + (−γt2)A2 + z
1
2
α4D
+
0 if Nmon = 0,
zα4+1C2 if Nmon ̸= 0, α4 ∈ N0,
−z−α4−1t2k̃12 C2 + z−α4tk̃12 D + z−α4+1E if Nmon ̸= 0, α4 ∈ Z<0,
F -manifold N2, with ∂2 ◦ ∂2 = 0,
E =
{
(t1 + c1)∂1 + γt2∂2 if α4 ̸= −1,
(t1 + c1)∂1 +
1
k̃1
tk̃1+1
2 ∂2 if α4 = −1
}
Euler field.
(iv) Generic type (Log): invariants k1 ∈ N, c1, ρ(1) ∈ C,
A2 = k1t
k1−1
2 C2,
B =
(
−t1 − c1 + zρ(1)
)
C1 − z
1
2
D, F -manifold N2, with ∂2 ◦ ∂2 = 0,
E = (t1 + c1)∂1 Euler field.
Theorem 8.1 is proved after Remark 8.2.
Remark 8.2. The other normal forms in Remark 6.6 for the generic type (Sem) with k2−k1 ∈ 2N
and for the generic type (Bra) give the following other normal forms. In both cases, the formulas
for A1 = C1, γ, B, the F -manifold and E are unchanged, only the matrix A2 changes. For the
generic type (Sem) with k2 − k1 ∈ 2N, A2 becomes
A2 = −γ−1
(
tk1−1
2 C2 + tk2−1
2 E
)
+ z
k2 − k1
2
ζt
(k2−k1−2)/2
2 E.
For the generic type (Bra), A2 becomes
A2 = −γ−1tk1−1
2 C2 + zk2t
k2−1
2 E.
Proof of Theorem 8.1. We prove the parts of Theorem 8.1 in the order (c), (d), (b), (a).
(c) Consider a rank 2 (TE)-structure
(
H [3] → C× (C, 0),∇[3]
)
(with coordinate t2 on (C, 0))
with trace free pole part and with logarithmic restriction over t2 = 0. If it admits an extension
to a pure (TLE)-structure, it is contained in Theorem 6.3. If not, then it is contained in Theo-
rem 6.7. The condition that the Higgs field is not vanishing, excludes the 8th and 9th cases in
Theorem 6.3 and the case (6.28) = case (III) in Theorem 6.7, see Remarks 6.8(ii) and (iii).
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 61
(d) Part (d) makes for such a (TE)-structure
(
H [3],∇[3]
)
the (TE)-structure (O(H),∇) =
pr∗2
(
O
(
H [3]
)
,∇[3]
)
⊗ E(t1+c1)/z explicit. The cooordinate t and the matrix A in Theorem 6.3
and in Remark 6.8(iv) become now t2 and A2. Here the matrices in the 6th case in Theorem 6.3
are not used, but the matrices in Remark 6.8(iv). The function f in Remark 6.8(iv) is now
specialized to f = t
k1/2
2 if α1 = α2 (⇒ case (II) and (6.31)) and to f = tk12 if α1 − α2 ∈ N
(case (I) and (6.30) or case (II) and (6.31)). The new matrix B is (−t1 − c1)C1 plus the
matrix B in Theorem 6.3 and in Remark 6.8(iv).
In the normal forms in Remark 6.8(iv) we replaced α1 and α2 by ρ(1) and α4 as follows
ρ(1) :=
α1 + α2 + 1
2
, α4 :=
{
α1 − α2 − 1 ∈ N0 in (6.30),
α2 − α1 − 1 ∈ Z<0 in (6.31).
(b) Now part (b) follows from inspection of the normal forms in part (d).
(a) Consider a (TE)-structure as in (a). Choose coordinates t = (t1, t2) on
(
M, t0
)
such that(
M, t0
)
=
(
C2, 0
)
and the germ of the F -manifold is in a normal form in Theorem 2.2 (especially
e = ∂1) and the Euler field has the form E = (t1 + c1)∂1 + g(t2)∂2 for some c1 ∈ C and some
g(t2) ∈ C{t2}.
Choose any C{t, z}-basis v of O(H)0 and consider its matrices A1, A2, B with z∇∂iv = vAi,
z2∇∂zv = vB. Now ∂1 = e implies A
(0)
1 = C1. We make a base change with the matrix
T ∈ GL2(C{t, z}) which is the unique solution of the differential equation
∂1T = −
(∑
k≥1
A
(k)
1 zk−1
)
T, T (z, 0, t2) = C1.
Then the matrices Ã1, Ã2, B̃ of the new basis ṽ = vT satisfy
Ã1 = C1, ∂1Ã2 = 0, ∂1B̃ = −C1, (8.2)
because (3.10) for i = 1 and (3.5) and (3.6) give
0 = z∂1T +A1T − TÃ1 = C1T − TÃ1 = T
(
C1 − Ã1
)
,
0 = z∂1Ã2 − z∂2Ã1 +
[
Ã1, Ã2
]
= z∂1Ã2,
0 = z∂1B − z2∂zÃ1 + zÃ1 +
[
Ã1, B̃
]
= z(∂1B + C1).
In Lemmata 3.10(c) and 3.11 we considered the (TE)-structure
(
O
(
H [2]
)
,∇[2]
)
= (O(H),∇)
⊗ Eρ(0)/z with trace free pole part. Here ρ(0) = −t1 − c1.
(8.2) shows that
(
H [2],∇[2]
)
is the pull back of its restriction
(
H [3],∇[3]
)
to ({0} × C, 0) ⊂(
C2, 0
)
. This and (O(H),∇) ∼=
(
O
(
H [2]
)
,∇[2]
)
⊗ E−ρ(0)/z in Lemma 3.10(c) imply (8.1). ■
Corollary 8.3. The following table gives for each logarithmic rank 2 (TE)-structure over
a point t0 its unfoldings within the set of (TE)-structures in Theorem 8.1(d). Here the set{
α0
1, α
0
2
}
⊂ C is the set of leading exponents in Theorem 4.20 of the logarithmic (TE)-structure
over t0. So, in the case Nmon = 0, α0
1 and α0
2 ∈ C are arbitrary. In the case Nmon ̸= 0, they
satisfy α0
1−α0
2 ∈ N0. Two conditions are c1 = 0 and ρ(1) =
α0
1+α0
2
2 . The other conditions and the
other invariants (though without their definition domains) are given in the table. All invariants
62 C. Hertling
in Theorem 8.1(d), which are not mentioned here, are (intended to be) arbitrary:
Generic type Invariants Nmon Condition
(Sem) : k2 > k1 k1, k2, ζ = 0 α0
1 − α0
2 = ±k1−k2
k1+k2
(Sem) : k2 = k1 k1, k2, α3 = 0 α0
1 − α0
2 = ±2α3
(Bra) k1, k2 = 0 α0
1 − α0
2 = ± −k2
k1+k2
(Reg) k1, α4 = 0 α0
1 − α0
2 = ±α4
(Reg) k̃1, α4 ̸= 0 α0
1 − α0
2 = |α4|
(Log) k1 = 0 α0
1 − α0
2 = ±1
(8.3)
Proof. This follows from inspection of the cases in Theorem 8.1(d). ■
Remark 8.4. Beware of the following:
(i) In the generic case (Sem) with k1 = k2 we have α3 ∈ R≥0∪H. Here α̃3 = −α3 is excluded,
as it gives an isomorphic unfolding.
(ii) In the generic cases (Reg) with α0
1−α0
2 ∈ C\{0} almost always α4 = α0
1−α0
2 and α̃4 = −α4
give (for the same k1 ∈ N respectively k̃1 ∈ N) two different unfoldings. The only exception
is the case Nmon = 0 and α0
1 − α0
2 = ±1, as then α4 = −1 is not allowed.
(iii) In the generic case (Log), one has one unfolding (and not two unfoldings) for each k1 ∈ N.
(iv) Unfoldings of generic type (Sem) with k2 > k1 and of generic type (Bra) exist only if
α0
1 − α0
2 ∈ (−1, 1) ∩Q∗ and Nmon = 0.
Theorem 8.5.
(a) Any unfolding of a rank 2 (TE)-structure of type (Log) over a point is induced by one in
the following subset of (TE)-structures in Theorem 8.1(d):
Generic type and invariants Condition
(Sem) : k2 − k1 > 0 odd gcd(k1, k2) = 1
(Sem) : k2 − k1 ∈ 2N, ζ = 0 gcd(k1, k2) = 1
(Sem) : k2 − k1 ∈ 2N, ζ ̸= 0 gcd
(
k1,
k1+k2
2
)
= 1
(Sem) : k2 = k1 k2 = k1 = 1
(Bra) gcd(k1, k2) = 1
(Reg) : Nmon = 0 k1 = 1
(Reg) : Nmon ̸= 0 k̃1 = 1
(Log) : Nmon = 0 k1 = 1
(8.4)
(b) The inducing (TE)-structure is not unique only if the original (TE)-structure has the
form φ∗(O(H [5]
)
,∇[5]
)
⊗ E−ρ(0)/z, where
(
H [5],∇[5]
)
is a logarithmic (TE)-structure over
a point t[5] and φ :
(
M, t0
)
→
{
t[5]
}
is the projection, and
(
H [5],∇[5]
)
is not one with
Nmon ̸= 0 and equal leading exponents α1 = α2. Then the original (TE)-structure is of
type (Log) everywhere with Higgs field endomorphisms CX ∈ O(M,t0)·id for any X ∈ T(M,t0).
(c) The (TE)-structures in the list in (a) are universal unfoldings of themselves.
The proof of Theorem 8.5 will be given in Section 8.3.
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 63
Theorem 8.6. The set of rank 2 (TE)-structures with primitive (not just generically primitive)
Higgs field over a germ
(
M, t0
)
of an F -manifold and with restriction of type (Log) over t0 is
(after the choice of suitable coordinates) the proper subset of those in the list (8.4) in Theorem 8.5
which satisfy k1 = 1 respectively k̃1 = 1. In the cases (Reg) and (Log), it coincides with the
list (8.4). In the cases (Sem) and (Bra), it is a proper subset.
Proof. The set of rank 2 (TE)-structures with primitive Higgs field over a germ
(
M, t0
)
of
an F -manifold and with restriction of type (Log) over t0 consists by Theorem 8.1(a)+(d) of
those (TE)-structures in Theorem 8.1(d) which satisfy A2(t2 = 0) /∈ C · C1. This holds if and
only if k1 = 1 respectively k̃1 = 1
(
k̃1 = 1 if the generic type is (Reg) and Nmon ̸= 0
)
, and then
A2(t2 = 0) ∈ {−γC2,−γD,C2}. Obviously, this is a proper subset of those in table (8.4) in the
generic cases (Sem) and (Bra), and it coincides with those in table (8.4) in the generic cases
(Reg) and (Log). ■
8.2 (TE)-structures over given F -manifolds with Euler fields
Remarks 8.7. For a given germ
((
M, t0
)
, ◦, e, E
)
of an F -manifold with Euler field, define
B1
((
M, t0
)
, ◦, e, E
)
:=
{
(TE)-structures over
(
M, t0
)
with generically primitive Higgs
field, inducing the given F -manifold structure with Euler field
}
,
B2
((
M, t0
)
, ◦, e, E
)
:= {(TE)-structures in B1 which are in table (8.4)},
B3
((
M, t0
)
, ◦, e, E
)
:= {(TE)-structures in B1 with primitive Higgs fields}.
Now we can answer the questions, how big these sets are. Often we write Bj instead of
Bj
((
M, t0
)
, ◦, e, E
)
, when the germ
((
M, t0
)
, ◦, e, E
)
is fixed.
(i) First we consider the cases when the germ
((
M, t0
)
, ◦, e, E
)
is regular. Compare Re-
mark 2.6(ii) and Remark 3.17(iii). By Malgrange’s unfolding result Theorem 3.16(c),
any (TE)-structure over
(
M, t0
)
is the universal unfolding of its restriction over t0, and
it is its own universal unfolding. So then B1 = B2 = B3, and the classification of the
(TE)-structures over points in Section 4 determines this space B1.
In the case of A2
1 with E = (u1 + c1)e1 + (u2 + c2)e2 with c1 ̸= c2, any (TE)-structure
over t0 is of type (Sem). Theorem 4.5 tells that then B1 is connected and 4-dimensional.
The parameters are the two regular singular exponents and two Stokes parameters.
In the case of N2 with E = (t1 + c1)∂1 + ∂2, any (TE)-structure over t0 is either of type
(Bra) or of type (Reg). Then B1 has one component for type (Bra) and countably many
components for type (Reg).
The component for type (Bra) is connected and 3-dimensional. The parameters are given
in Theorem 4.11, they are ρ(1), δ(1) and Eig(Mmon)
(
here ρ(0)
(
t0
)
= −c1 is fixed, and one
eigenvalue and ρ(1) determine the other eigenvalue
)
.
Corollary 4.18 gives the countably many components for type (Reg). One is 1-dimensional,
the others are 2-dimensional.
(ii) Now we consider the cases when the germ
((
M, t0
)
, ◦, e, E
)
is not regular. Then E|t0 =
c1∂1 for some c1 ∈ C. If (O(H),∇) is a (TE)-structure in Bj
((
M, t0
)
, ◦, e, E
)
, then
(O(H),∇) ⊗ E−c1/z is a (TE)-structure in Bj
((
M, t0
)
, ◦, e, E − c1∂1
)
. Therefore we can
and will restrict to the cases with E|t0 = 0.
Theorem 8.1(d) gives the (TE)-structures in B1, Theorem 8.5(a) gives the (TE)-structures
in B2, and Theorem 8.6 gives the (TE)-structures in B3. For each germ
((
M, t0
)
, ◦, e, E
)
64 C. Hertling
with E|t0 = 0
B1 ⊃ B2 ⊃ B3.
In the cases A2
1 and I2(m), the Euler field with E|t0 = 0 is unique on
(
M, t0
)
, therefore
we do not write it down.
In the case of I2(m) with m ∈ 2N (this includes the case A2
1 = I2(2))
B1(I2(m)) ∼=
⋃̇
(k1,k2)∈N2 : k1+k2=m,k2≥k1
C2,
B2(I2(m)) ∼=
⋃̇
(k1,k2)∈N2 : k1+k2=m,k2≥k1,gcd(k1,m/2)=1
C2,
B3(I2(m)) ∼= C2, here (k1, k2) = (1,m− 1). (8.5)
The 2 continuous parameters are the regular singular exponents of the (TE)-structures at
generic points in M .
In the case of I2(m) with m ≥ 3 odd,
B1(I2(m)) ∼=
⋃̇
(k1,k2)∈N2 : k1+k2=m,k2>k1
C,
B2(I2(m)) ∼=
⋃̇
(k1,k2)∈N2 : k1+k2=m,k2>k1,gcd(k1,k2)=1
C,
B3(I2(m)) ∼= C, here (k1, k2) = (1,m− 1). (8.6)
For odd m ≥ 3, the regular singular exponents of the (TE)-structures at generic points
in M coincide and give the continuous parameter.
Especially, for m ∈ {2, 3}
B1(I2(m)) = B2(I2(m)) = B3(I2(m)) ∼=
{
C2 if m = 2,
C if m = 3.
The F -manifold N2 allows by Theorem 2.3 many nonisomorphic Euler fields with E|t0 = 0,
the cases (2.5)–(2.7) with c1 = 0.
The case (2.5), E = t1∂1: Here each (TE)-structure has generic type (Log) and semisimple
monodromy. Here
B1(N2, E) ∼=
⋃̇
k1∈N
C,
B2(N2, E) = B3(N2, E) ∼= C, here k1 = 1.
The continuous parameter is ρ(1) in Theorem 8.1(d) (iv) or, equivalently, one of the two
residue eigenvalues
(
which are ρ(1) ± 1
2
)
.
The case (2.7), E = t1∂1 + tr2
(
1 + c3t
r−1
2
)
for some r ∈ Z≥2 and some c3 ∈ C: Here each
(TE)-structure has generic type (Reg) and satisfies Nmon ̸= 0. Here
B1(N2, E)
{
= ∅, if c3 ∈ C∗,
∼= C if c3 = 0,
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 65
B2(N2, E) = B3(N2, E)
{
= ∅, if c3 ∈ C∗ or r ≥ 3,
= B1(N2, E) if c3 = 0 and r = 2.
So, (N2, E) with c3 ∈ C∗ does not allow (TE)-structures over it, and (N2, E) with c3 = 0
and r ≥ 3 does not allow (TE)-structures over it with primitive Higgs field. If Bj(N2, E)
̸= ∅ then Bj(N2, E) ∼= C and the continuous parameter is ρ(1) in Theorem 8.1(d) (iii).
The case (2.6), E = t1∂1+c2t2∂2 for some c2 ∈ C∗: This is a rich case. Here we decompose
Bj = Bj(N2, E) as
Bj = B
(Reg),0
j ∪̇ B
(Reg), ̸=0
j ∪̇ B
(Bra)
j ,
where the first set contains (TE)-structures of generic type (Reg) with Nmon = 0, the
second set contains (TE)-structures of generic type (Reg) with Nmon ̸= 0, and the third
set contains (TE)-structures of generic type (Bra). Then
B
(Reg),0
1
∼=
⋃̇
k1∈N
C, B
(Reg),0
2 = B
(Reg),0
3
∼= C,
B
(Reg), ̸=0
1
{
= ∅ if c2 ∈ C∗ \Q∗,
∼=
⋃̇
(k1,α4)∈N×Z : k1c2=1+α4
C if c2 ∈ Q∗,
B
(Reg), ̸=0
2 = B
(Reg), ̸=0
3
{
= ∅ if c2 ∈ C \ Z,
∼= C if c2 ∈ Z \ {0},
B
(Bra)
1 = B
(Bra)
2 = B
(Bra)
3 = ∅ if c−1
2 ∈ C∗ \ Z≥2,
B
(Bra)
1
∼=
⋃̇
(k1,k2)∈N2 : k1+k2=c−1
2
C,
B
(Bra)
2
∼=
⋃̇
(k1,k2)∈N2 : k1+k2=c−1
2 ,gcd(k1,k2)=1C,
B
(Bra)
3
∼= C, here (k1, k2) =
(
1, c−1
2 − 1
)
if c−1
2 ∈ Z≥2.
Remarks 8.8.
(i) Theorem 8.1(d) (i) tells how many (TE)-structures exist over the F -manifold with Euler
field I2(m), such that the Higgs bundle is generically primitive and induces this F -manifold
structure. There are
[
m
2
]
many holomorphic families from the different choices of (k1, k2)
∈ N2 with k2 ≥ k1 and k1+k2 = m. They have 2 parameters if m is even and 1 parameter
if m is odd, compare (8.5) and (8.6). For each I2(m), only one of these families consists
of (TE)-structures with primitive Higgs fields.
(ii) Consider m ≥ 3. Write M = C2 for the F -manifold I2(m) in Theorem 2.2, and M [log] =
C × {0} for the subset of points where the multiplication is not semisimple. Over these
points the restricted (TE)-structures are of type (Log). We checked that there are
[
m
2
]
many Stokes structures which give (TE)-structures on M \M [log]. Because of (i), all these
(TE)-structures extend holomorphically over M [log], and they give the
[
m
2
]
holomorphic
families of (TE)-structures on I2(m) in (i).
(iii) Especially remarkable is the case A2
1 = I2(2). There Theorem 8.1(a)+(d) (i) implies
directly that each holomorphic (TE)-structure over A2
1 with generically primitive Higgs
field has primitive Higgs field and is an elementary model (Definition 4.4), so it has trivial
Stokes structure.
(iv) This result is related to much more general work in [3] and [22] on meromorphic connections
over the F -manifold An
1 near points where some of the canonical coordinates coincide. Let
66 C. Hertling
us restrict to the special case of a neighborhood of a point where all canonical coordinates
coincide. This generalizes the germ at 0 of A2
1 to the germ at 0 of An
1 .
[3, Theorem 1.1] and [22, Theorem 3] both give the triviality of the Stokes structure.
Though their starting points are slightly restrictive. [3] starts in our notation from pure
(TLE)-structures with primitive Higgs fields. The step before in the case of A2
1, passing
from a (TE)-structure over A2
1 to a pure (TLE)-structure, is done essentially in our Theo-
rem 6.2 (a) (iii). Our argument for the triviality of the Stokes structure is then contained
in the proof of Theorem 6.3.
[22] starts in our notation from (TE)-structures which are already formally isomorphic to
sums
⊕n
i=1 Eui/zzαi . Then it is shown that they are also holomorphically isomorphic to
such sums. In this special case, Corollary 5.7 in [7] give this implication, too.
(v) In (ii) we stated that in the case of I2(m) with m ≥ 3, each (TE)-structure on M \M [log]
with primitive Higgs field extends holomorphically to M . In the case of N2 this does not
hold in general. For example, start with the flat rank 2 bundle H ′ → C∗ ×M , where
M = C2 (with coordinates t = (t1, t2)) with semisimple monodromy with two different
eigenvalues λ1 and λ2. Choose α1, α2 ∈ C with e−2πiαj = λj . Let sj ∈ Cαj be generating
elementary sections. Define the new basis
v = (v1, v2) =
(
et1/z
(
s1 + e−1/t2s2
)
, et1/z(zs2)
)
on H ′|M ′ , where M ′ :=M \ C× {0}. Then
z∇∂1v = v · C1,
z∇∂2v = v · t−2
2 e−1/t2C2,
z2∇∂zv = v ·
(
−t1C1 + (α2 − α1)e
−1/t2C2 + z
(
α1 0
0 α2 + 1
))
.
So, we obtain a regular singular (TE)-structure on M ′ with primitive Higgs field. The
F -manifold structure on M ′ is given by e = ∂1 and ∂2 ◦ ∂2 = 0, so it is N2, and the Euler
field is E = t1∂1 + (α1 − α2)t
2
2∂2. F -manifold and Euler field extend from M ′ to M , but
not the (TE)-structure.
8.3 Proof of Theorem 8.5
(a) Let
(
H → C ×
(
M, t0
)
,∇
)
be an unfolding of a (TE)-structure of type (Log) over t0. The
(TE)-structure
(
H [2] → C×
(
M, t0
)
,∇[2]
)
in Lemma 3.10(c) with
(
O
(
H [2]
)
,∇[2]
)
= (O(H),∇)⊗
E−ρ(0)/z has trace free pole part. Lemma 3.10(d) and (e) apply. Because of them, it is sufficient
to prove that the (TE)-structure
(
H [2],∇[2]
)
is induced by a (TE)-structure
(
H [3] → C ×(
M [3], t[3]
)
,∇[3]
)
over
(
M [3], t[3]
)
= (C, 0) via a map φ :
(
M, t0
)
→
(
M [3], t[3]
)
, where the (TE)-
structure
(
H [3],∇[3]
)
is one of the (TE)-structures in the 1st to 7th cases in Theorem 6.3 or one
of the (TE)-structures in the cases (I) or (II) in Theorem 6.7 with invariants as in table (8.4).
Then the (TE)-structure
(
H [4],∇[4]
)
which is constructed in Lemma 3.10(d) from
(
H [3],∇[3]
)
is one of the (TE)-structures in Theorem 8.1(d) with invariants as in table (8.4), and it induces
by Lemma 3.10(e) the (TE)-structure (H,∇).
From now on we suppose ρ(0) = 0, so (H,∇) =
(
H [2],∇[2]
)
. We consider the invariants
δ(0), δ(1) ∈ OM,t0 and U and the four possible generic types (Sem), (Bra), (Reg) and (Log),
which are defined by the following table, analogously to Definition 6.1,
(Sem) (Bra) (Reg) (Log)
δ(0) ̸= 0 δ(0) = 0, δ(1) ̸= 0 δ(0) = δ(1) = 0, U ̸= 0 U = 0
First we treat the generic types (Reg) and (Log), then the generic type (Sem) and (Bra).
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 67
Generic types (Reg) and (Log). Then the (TE)-structure (H,∇) is regular singular. We can use
the results in Section 7 (which built on Theorems 6.3 and 6.7). Choose a marking for the (TE)-
structure (H,∇). Then by Remark 7.5(i), there is a unique map φ :
(
M, t0
)
→M (Href,∞,Mref),reg
which maps t ∈ M to the point in the moduli space over which one has up to isomorphism
the same marked (TE)-structure as over t. The map φ is holomorphic. By Remark 7.5(i)+(ii)
it maps
(
M, t0
)
to one projective curve which is isomorphic to M (3),0,α1,α2 or M (3),̸=0,α1,α2 or
M (3),0,α1,log. The (TE)-structure (H,∇) is induced by the (TE)-structure over this curve via
the map φ. The point t0 is mapped to 0 or ∞ in the cases M (3),0,α1,α2 or M (3),̸=0,α1,α2
(
not 0
in the case M (3), ̸=0,α1,α1
)
as the (TE)-structure over t0 is logarithmic. The germs at 0 and ∞
in M (3),0,α1,α2 and M (3), ̸=0,α1,α2
(
not 0 in the case M (3),̸=0,α1,α1
)
and the germ at any point t
(3)
2
in M (3),0,α1,log are contained in table (8.4). This shows Theorem 8.5 for the generic cases (Reg)
and (Log).
Generic types (Sem) and (Bra). We choose a (connected and sufficiently small) representativeM
of the germ
(
M, t0
)
, and we choose on it coordinates t = (t1, . . . , tm) (with m = dimM) with
t0 = 0. We denote by M [log] the analytic hypersurface
M [log] :=
(
δ(0)
)−1
(0), if the generic type is (Sem),(
δ(1)
)−1
(0), if the generic type is (Bra).
It contains t0. Choose a disk ∆ ⊂ M through t0 with ∆ \ {t0} ⊂ M \M [log]. The restricted
(TE)-structure (H,∇)|C×(∆,t0) has the same generic type as the (TE)-structure (H,∇). The
restricted (TE)-structure (H,∇)|C×(∆,t0) is isomorphic to a (TE)-structure in the cases 1, 2, 3
or 4 in Theorem 6.3.
The parameters of the restricted (TE)-structure (H,∇)|C×(∆,t0) are given in the following
table:
Generic type Parameters
(Sem) k1, k2 ∈ N with k2 ≥ k1, ρ
(1) ∈ C,{
ζ ∈ C if k2 − k1 ∈ 2N,
α3 ∈ R≥0 ∪H if k1 = k2
(Bra) k1, k2 ∈ N, ρ(1) ∈ C
There is a unique pair
(
k01, k
0
2
)
∈ N2 with (k1, k2) ∈ Q>0 ·
(
k01, k
0
2
)
and with the conditions in
table (8.7),
Generic type and invariants Conditions
(Sem): k2 − k1 > 0 odd gcd
(
k01, k
0
2
)
= 1
(Sem): k2 − k1 ∈ 2N, ζ = 0 gcd
(
k01, k
0
2
)
= 1
(Sem): k2 − k1 ∈ 2N, ζ ̸= 0 k02 − k01 ∈ 2N, gcd
(
k01,
k01+k02
2
)
= 1
(Sem): k2 = k1 k02 = k01 = 1
(Bra) gcd
(
k01, k
0
2
)
= 1
(8.7)
In fact, it is the pair
(
k01, k
0
2
)
∈ N2 of minimal numbers which satisfies
(k1, k2) ∈ N ·
(
k01, k
0
2
)
and which satisfies in the case (Sem) with k2 − k1 ∈ 2N and ζ ̸= 0 additionally k02 − k01 ∈ 2N.
68 C. Hertling
We denote by
(
H [3] → C ×
(
M [3], t[3]
)
,∇[3]
)
the (TE)-structure over
(
M [3], t[3]
)
= (C, 0)
which has
(
k01, k
0
2
)
instead of (k1, k2), but which has the same other parameters as the restricted
(TE)-structure (H,∇)|C×(∆,t0).
We have seen in Remarks 6.5(ii) and (iii) that the restricted (TE)-structure (H,∇)|C×(∆,t0) is
induced by the (TE)-structure
(
H [3],∇[3]
)
via the branched covering φ∆ : (∆, t0) →
(
M [3], t[3]
)
with φ∆(τ) = τk1/k
0
1 . Here τ denotes that coordinate on ∆ with which (H,∇)|C×(∆,t0) can be
brought to a normal form in the cases 1, 2, 3 and 4 in Theorem 6.3.
It rests to extend φ∆ to a map φ : M →M [3] such that (H,∇) is induced by
(
H [3],∇[3]
)
via
this map φ.
Claim 8.9. There exists a unique holomorphic function φ ∈ OM with
φ|∆ = φ∆, (8.8)
δ(0) = −φk01+k02 if the generic type is (Sem), (8.9)
δ(1) =
k02
k01 + k02
· φk01+k02 if the generic type is (Bra). (8.10)
Proof. Choose any point t[1] ∈ M [log] and any disk ∆[1] through t[1] with ∆[1] \
{
t[1]
}
⊂
M \M [log]. In order to show the existence of a function φ ∈ OM with (8.9) respectively (8.10),
it is sufficient to show that δ(0)|∆[1] respectively δ(1)|∆[1] has at t[1] a zero of an order which is
a multiple of k01 + k02.
The restricted (TE)-structure (H,∇)|C×(∆[1],t[1]) has the same generic type as (H,∇) and is
isomorphic to a (TE)-structure in the cases 1, 2, 3 or 4 in Theorem 6.3. Its invariants k1 and k2
are here called k
[1]
1 and k
[1]
2 , in order to distinguish them from the invariants of (H,∇)|(∆,t0).
We want to show(
k
[1]
1 , k
[2]
2
)
∈ N ·
(
k01, k
0
2
)
. (8.11)
We did not say much about the Stokes structure. Here we need the following properties of
it, if the generic type is (Sem):
k2 = k1
(1)⇐⇒ (H,∇)|C×{t[2]} has trivial Stokes structure for t[2] ∈ ∆ \
{
t0
}
(2)⇐⇒ (H,∇)|C×{t[2]} has trivial Stokes structure for t[2] ∈ ∆[1] \
{
t[1]
}
(3)⇐⇒ k
[1]
2 = k
[1]
1 .
(1)
=⇒ and
(3)⇐= are obvious from the normal form in the 3rd case in Theorem 6.3. It is not hard
to see that the normal forms for fixed t ∈ C∗ in the 1st and 2nd case in Theorem 6.3 are not
holomorphically isomorphic to an elementary model in Definition 4.4 (see also Remark 8.8(ii)).
This shows
(1)⇐= and
(3)
=⇒. The equivalence
(2)⇐⇒ is a consequence of the invariance of the Stokes
structure within isomonodromic deformations.
In the generic type (Sem) with k1 = k2 we have also k
[1]
2 = k
[1]
1 and k02 = k01 = 1, and thus
especially (8.11).
Now consider the cases with k2 > k1. This comprises the generic type (Bra) and gives in the
generic type (Sem) also k02 > k01 and k
[1]
2 > k
[1]
1 . So (H,∇)|C×(∆[1],t[1]) is in the 1st, 2nd or 4th
case in Theorem 6.3. The number b
(1)
3 in Theorem 6.3 is uniquely determined by the properties
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 69
b
(1)
3 ∈ Q ∩
]−1
2 , 0
[
and Eig(Mmon) =
{
exp
(
−2πi(ρ(1)±b(1)3
))}
(see Remark 6.4(i) for the second
property). Therefore
k01 − k02
2
(
k01 + k02
) =
k1 − k2
2(k1 + k2)
= b
(1)
3 =
k
[1]
1 − k
[1]
2
2
(
k
[1]
1 + k
[1]
2
) in the case (Sem),
−k02
2
(
k01 + k02
) =
−k2
2(k1 + k2)
= b
(1)
3 =
−k[1]2
2
(
k
[1]
1 + k
[1]
2
) in the case (Bra). (8.12)
This implies
(
k
[1]
1 , k
[1]
2
)
∈ Q>0 ·
(
k01, k
0
2
)
. In the cases with gcd
(
k01, k
0
2
)
= 1 (8.11) follows.
If gcd
(
k01, k
0
2
)
̸= 1, then the generic type is (Sem), k2 − k1 ∈ 2N, k02 − k01 ∈ 2N, and the
invariant ζ of (H,∇)|C×(∆,t0) is ζ ̸= 0. However, then the regular singular exponents α1 and α2
of the restriction of the (TE)-structure (H,∇) over points in M \M [log] are invariants of the
(TE)-structure (H,∇). By (6.11) and (8.12) also ζ is an invariant of the (TE)-structure (H,∇).
Now ζ ̸= 0 implies k
[1]
2 − k
[1]
1 ∈ 2N. Again (8.11) follows.
Equations (6.8) and (6.12) imply that δ(0)|∆[1] respectively δ(1)|∆[1] has at t[1] a zero of an
order which is a multiple of k01 + k
0
2. Therefore a function φ ∈ OM with (8.9) respectively (8.10)
exists.
Equations (6.8) and (6.12) for (H,∇)|C×(∆,t0) tell
δ(0)|∆ = −τk1+k2 = −
(
τk1/k
0
1
)k01+k02 = −
(
φ∆
)k01+k02 in the case (Sem),
δ(1)|∆ =
k2
k1 + k2
τk1+k2 =
k02
k01 + k02
(
τk1/k
0
1
)k01+k02 =
k02
k01 + k02
(
φ∆
)k01+k02 in the case (Bra).
Therefore a function φ as in the claim exists and is unique. ■
Now compare the (TE)-structures (H,∇) and φ∗(H [3],∇[3]
)
over M . Both extend to pure
(TLE)-structures. For φ∗(H [3],∇[3]
)
, one uses the pull back φ∗(v[3]) of the basis v[3] which
gives for
(
H [3],∇[3]
)
the Birkhoff normal form in Theorem 6.3. For (H,∇) one starts with
the analogous basis v∆ for H|C×∆ which gives for (H,∇)|C×∆ the Birkhoff normal form in
Theorem 6.3. It has a unique extension v to C ×M which still yields a Birkhoff normal form.
Compare Remark 3.19(ii) for this.
Remarks 6.5(ii) and (iii) (or simply the Birkhoff normal forms in Theorem 6.3) tell that the
map
(
φ∗v[3]
)
|C×∆ 7→ v∆ = v|C×∆ is an isomorphism of pure (TLE)-structures.
Now consider a point t[2] ∈ ∆ \
{
t0
}
and its image t[4] := φ
(
t[2]
)
∈ M [3] \
{
t[3]
}
= C \ {0}.
Over the germ
(
M [3], t[4]
)
, the (TE)-structure
(
H [3],∇[3]
)
is the part with trace free pole part
of a universal unfolding of
(
H [3],∇[3]
)
|C×{t[4]}. Therefore in a neighborhood U ⊂ M of t[2],
the (TE)-structure (H,∇)|C×U is induced by
(
H [3],∇[3]
)
|C×(M [3],t[4]) via a map φ̃ : U → M [3].
We can choose it such that
φ̃|∆ = φ∆. (8.13)
Equations (6.8) and (6.12) tell
δ(0)|U = −(φ̃)k
0
1+k02 in the case (Sem), (8.14)
δ(1)|U =
k02
k01 + k02
(φ̃)k
0
1+k02 in the case (Bra). (8.15)
Equations (8.13)–(8.15) and Claim 8.9 imply φ̃ = φ|U . Therefore the matrices in Birkhoff normal
form for the basis v of (H,∇) coincide on C× U with the matrices in Birkhoff normal form for
70 C. Hertling
the basis φ∗(v[3]) of φ∗(H [3],∇[3]
)
. As all matrices are holomorphic on C ×M , they coincide
pairwise on C ×M . Therefore the pure (TLE)-structure (H,∇) with basis v is isomorphic to
the pure (TLE)-structure φ∗(H [3],∇[3]
)
with basis φ∗(v[3]). This finishes the proof of part (a)
of Theorem 8.5.
(b) If the original (TE)-structure (H,∇) has the form φ∗(O(H [5]),∇[5]
)
⊗ E−ρ(0)/z then the
(TE)-structure
(
H [2],∇[2]
)
with trace free pole part which was associated to (H,∇) at the
beginning of the proof of part (a), has the form φ∗(O(H [5]),∇[5]
)
.
Then any (TE)-structure
(
H [3],∇[3]
)
over
(
M [3], t[3]
)
= (C, 0) works, whose restriction
over t[3] is the given logarithmic (TE)-structure
(
H [5],∇[5]
)
.
In the cases with Nmon = 0, table (8.4) offers one of generic type (Sem) with k1 = k2 = 1
(and some with k2 > k1 if α1 − α2 ∈ Q ∩ (−1, 1)) and one or two of generic type (Reg), see
table (8.3). In the cases with Nmon ̸= 0, table (8.4) offers two of generic type (Reg) if the
leading exponents α1 and α2 satisfy α1 − α2 ∈ N, and one if they satisfy α1 = α2, compare also
Figures 7.4 and 7.5 in Theorem 7.4(b). Therefore the inducing (TE)-structure in table (8.4) is
not unique except for the case Nmon ̸= 0 and α1 = α2, if the original (TE)-structure has the
form φ∗(O(H [5]),∇[5]
)
⊗ E−ρ(0)/z.
In the other cases, the proof of part (a) shows the uniqueness of the (TE)-structure(
H [3],∇[3]
)
. The uniqueness of
(
H [3],∇[3]
)
gives also the uniqueness of
(
H [4],∇[4]
)
in the first
paragraph of the proof of part (a).
(c) This follows from the uniqueness in part (b). ■
9 A family of rank 3 (TE)-structures
with a functional parameter
M. Saito presents in [23] a family of Gauss–Manin connections with a functional parameter.
In the arXiv paper [23], the bundle has rank n, but in a preliminary version it has rank 3 and
is more transparent.
Here we translate the rank 3 example by a Fourier–Laplace transformation into a family of
(TE)-structures with primitive Higgs fields over a fixed 3-dimensional globally irreducible F -
manifold with an Euler field, such that the F -manifold with Euler field is nowhere regular. The
family of (TE)-structures has a functional parameter h(t2) ∈ C{t2}.
In the following, we write down a (TE)-structure of rank 3 on a manifold M = C3 with
coordinates t1, t2, t3. The restriction to
{
t ∈ C3 | t1 = 0
}
= {0} × C2 is a FL-transformation
of Saito’s example. The parameter t1 and this F -manifold are not considered in [23]. There the
base space has only the two parameters t2 and t3. Choose an arbitrary function h(t2) ∈ C{t2}
with h′′(0) ̸= 0.
LetH ′ → C∗×M be a holomorphic vector bundle with flat connection with trivial monodromy
and basis of global flat sections s1, s2, s3. Define an extension to a vector bundle H → C×M
using the following holomorphic sections of H ′, which also form a basis of sections of H ′:
v1 := et1/z ·
(
zs1 + t2 · zs2 + h(t2) · zs3 + t3 · z2s3
)
,
v2 := et1/z ·
(
z2s2 + h′(t2) · z2s3
)
, v3 := et1/z · z3s3.
Denote v := (v1, v2, v3). Denote ∂tj := ∂j . Then
z∇∂1v = v · 13,
z∇∂2v = v ·
0 0 0
1 0 0
0 h′′(t2) 0
,
Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 71
z∇∂3v = v ·
0 0 0
0 0 0
1 0 0
,
z2∂zv = v ·
−t1 · 13 +
0 0 0
0 0 0
t3 0 0
+ z · v ·
1 0 0
0 2 0
0 0 3
.
Write ∂ := (∂1, ∂2, ∂3). The pole parts give the multiplication ◦ on the F -manifold and the
Euler field E by
∂1 ◦ ∂ = ∂ · 13,
∂2 ◦ ∂ = ∂ ·
0 0 0
1 0 0
0 h′′(t2) 0
,
∂3 ◦ ∂ = ∂ ·
0 0 0
0 0 0
1 0 0
,
E ◦ ∂ = −∂ ·
−t1 · 13 +
0 0 0
0 0 0
t3 0 0
, so E = t1 · ∂1 − t3 · ∂3.
One can introduce a new coordinate system
(
t̃1, t̃2, t̃3
)
=
(
t1, t̃2, t3
)
on the germ (M, 0) with
∂t̃2 =
1√
h′′(t2)
· ∂2.
Denote ∂̃j := ∂t̃j and ∂̃ :=
(
∂̃1, ∂̃2, ∂̃3
)
=
(
∂1, ∂̃2, ∂3
)
. Introduce also the new section
ṽ2 :=
1√
h′′(t2)
· v2,
and the new basis ṽ =
(
ṽ1, ṽ2, ṽ3
)
=
(
v1, ṽ2, v3
)
of the given (TE)-structure. Then
z∇
∂̃1
ṽ = ṽ · 13,
z∇
∂̃2
ṽ = ṽ ·
0 0 0
1 0 0
0 1 0
+ ṽ ·
0 0 0
0 ∂2
1√
h′′(t2)
0
0 0 0
,
z∇
∂̃3
ṽ = ṽ ·
0 0 0
0 0 0
1 0 0
,
z2∂z ṽ = ṽ ·
−t1 · 13 +
0 0 0
0 0 0
t3 0 0
+ z · ṽ ·
1 0 0
0 2 0
0 0 3
.
In the new coordinates the multiplication becomes simpler and independent of the choice of h(t2)
(as long as h′′(t2) ̸= 0):
∂̃1 ◦ ∂̃ = ∂̃ · 13,
∂̃2 ◦ ∂̃ = ∂̃ ·
0 0 0
1 0 0
0 1 0
,
72 C. Hertling
∂̃3 ◦ ∂̃ = ∂̃ ·
0 0 0
0 0 0
1 0 0
,
E ◦ ∂̃ = −∂̃ ·
−t1 · 13 +
0 0 0
0 0 0
t3 0 0
,
so E = t1 · ∂̃1 − t3 · ∂̃3.
This is the nilpotent F -manifold for n = 3 in [4, Theorem 3]. However, the Euler field here is
different from the one in [4, Theorem 3]. The endomorphism E◦ here is not regular, but has
only the one eigenvalue t1 and has for t3 ̸= 0 one Jordan block of size 2 × 2 and one Jordan
block of size 1× 1 and is semisimple for t3 = 0.
The sections v1, v2, v3 define also an extension Ĥ → P1 such that the (TE)-structure extends
to a pure (TLE)-structure.
Furthermore v satisfies all properties of the section ζ in Theorem 6.6(b) in [7]. Thus the
F -manifold with Euler field is enriched to a flat F -manifold with Euler field (Definition 3.1(b)
in [7]).
If we try to introduce a pairing which would make it into a pure (TLEP )-structure, we get
a constraint h′′(t2) = const. However, probably similar higher dimensional examples allow also
an extension to pure (TLEP )-structures while keeping the functional freedom. This would give
families of Frobenius manifolds with Euler fields with functional freedom on a fixed F -manifold
with Euler field.
In the example above, t1, t2, t3 are flat coordinates and t̃1 = t1, t̃2, t̃3 = t3 are generalized
canonical coordinates (in which the multiplication has simple formulas).
Acknowledgements
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foun-
dation) – 242588615. I would like to thank Liana David for a lot of joint work on (TE)-structures.
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1 Introduction
2 The two-dimensional F-manifolds and their Euler fields
3 (TE)-structures in general
3.1 Definitions
3.2 (TE)-structures with trace free pole part
3.3 (TE)-structures over F-manifolds with Euler fields
3.4 Birkhoff normal form
3.5 Regular singular (TE)-structures
3.6 Marked (TE)-structures and moduli spaces for them
4 Rank 2 (TE)-structures over a point
4.1 Separation into 4 cases
4.2 The case (Sem)
4.3 Joint considerations on the cases (Bra), (Reg) and (Log)
4.4 The case (Bra)
4.5 The case (Reg) with `3́9`42`"̇613A``45`47`"603AtrU=0
4.6 The case (Log) with `3́9`42`"̇613A``45`47`"603AtrU=0
5 Rank 2 (TE)-structures over germs of regular F-manifolds
6 1-parameter unfoldings of logarithmic (TE)-structuresover a point
6.1 Numerical invariants for such (TE)-structures
6.2 1-parameter unfoldings with trace free pole part of logarithmic pure (TLE)-structures over a point
6.3 Generically regular singular (TE)-structures over (C,0) with logarithmic restriction over t0=0 and not semisimple monodromy
7 Marked regular singular rank 2 (TE)-structures
8 Unfoldings of rank 2 (TE)-structures of type (Log) over a point
8.1 Classification results
8.2 (TE)-structures over given F-manifolds with Euler fields
8.3 Proof of Theorem 8.5
9 A family of rank 3 (TE)-structures with a functional parameter
References
|
| id | nasplib_isofts_kiev_ua-123456789-211445 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T12:06:11Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Hertling, Claus 2026-01-02T08:35:38Z 2021 Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1. Claus Hertling. SIGMA 17 (2021), 082, 73 pages 1815-0659 2020 Mathematics Subject Classification: 34M56; 34M35; 53C07 arXiv:2009.14314 https://nasplib.isofts.kiev.ua/handle/123456789/211445 https://doi.org/10.3842/SIGMA.2021.082 Holomorphic vector bundles on ℂ × , a complex manifold, with meromorphic connections with poles of Poincaré rank 1 along {0} × , arise naturally in algebraic geometry. They are called ( )-structures here. This paper takes an abstract point of view. It gives a complete classification of all ( )-structures of rank 2 over germs ( , ⁰) of manifolds. In the case of , they separate into four types. Those of the three types have universal unfoldings; those of the fourth type (the logarithmic type) do not. The classification of unfoldings of ( )-structures of the fourth type is rich and interesting. The paper finds and lists all ( )-structures which are basic in the following sense: Together they induce all rank 2 ( )-structures, and each of them is not induced by any other ( )-structure in the list. Their base spaces turn out to be 2-dimensional F-manifolds with Euler fields. The paper also provides a classification of all rank 2 ( )-structures over it. Also, this classification is surprisingly rich. The backbone of the paper is normal forms. Though also the monodromy and the geometry of the induced Higgs fields and of the base spaces are important and are considered. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 242588615. I would like to thank Liana David for a lot of joint work on (TE)-structures. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 Article published earlier |
| spellingShingle | Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 Hertling, Claus |
| title | Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 |
| title_full | Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 |
| title_fullStr | Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 |
| title_full_unstemmed | Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 |
| title_short | Rank 2 Bundles with Meromorphic Connections with Poles of Poincaré Rank 1 |
| title_sort | rank 2 bundles with meromorphic connections with poles of poincaré rank 1 |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211445 |
| work_keys_str_mv | AT hertlingclaus rank2bundleswithmeromorphicconnectionswithpolesofpoincarerank1 |