Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds
We study mirror symmetry of complete intersection Calabi-Yau manifolds that have birational automorphisms of infinite order. We observe that movable cones in birational geometry are transformed, under mirror symmetry, to the monodromy nilpotent cones, which are naturally glued together.
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| Cite this: | Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds / S. Hosono, H. Takagi // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 46 назв. — англ. |
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| citation_txt | Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds / S. Hosono, H. Takagi // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 46 назв. — англ. |
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| description | We study mirror symmetry of complete intersection Calabi-Yau manifolds that have birational automorphisms of infinite order. We observe that movable cones in birational geometry are transformed, under mirror symmetry, to the monodromy nilpotent cones, which are naturally glued together.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 039, 37 pages
Movable vs Monodromy Nilpotent Cones
of Calabi–Yau Manifolds
Shinobu HOSONO and Hiromichi TAKAGI
Department of Mathematics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588, Japan
E-mail: hosono@math.gakushuin.ac.jp, hiromici@math.gakushuin.ac.jp
Received September 11, 2017, in final form April 23, 2018; Published online May 02, 2018
https://doi.org/10.3842/SIGMA.2018.039
Abstract. We study mirror symmetry of complete intersection Calabi–Yau manifolds which
have birational automorphisms of infinite order. We observe that movable cones in birational
geometry are transformed, under mirror symmetry, to the monodromy nilpotent cones which
are naturally glued together.
Key words: Calabi–Yau manifolds; mirror symmetry; birational geometry; Hodge theory
2010 Mathematics Subject Classification: 14E05; 14E07; 14J33; 14N33
1 Introduction
A smooth projective variety X of dimension n is called a Calabi–Yau n-fold if the canonical
bundle KX is trivial and H i(X,OX) = 0, 1 ≤ i ≤ n−1. In the 90’s, the idea of mirror symmetry
was discovered in theoretical physics and has long been a source of many mathematical ideas
related to Calabi–Yau manifolds. After more than 20 years since its discovery, we have now
several approaches [17, 18, 34, 45] toward mathematical understanding of the symmetry.
In this paper, we will focus on “classical” mirror symmetry of Calabi–Yau threefolds, i.e.,
we compare two different moduli spaces associated to Calabi–Yau threefolds, the Kähler moduli
and the complex structure moduli spaces, considering Calabi–Yau threefolds which have several
birational models. According to birational geometry of higher dimensional manifolds, if a Calabi–
Yau threefold X has birational models, then the Kähler cone of X can be extended to the
movable cone Mov(X) [32, 39]. On the mirror side, corresponding to each birational model, there
appears a special boundary point called large complex structure limit, which is characterized by
unipotent monodromy [38]. Using this unipotent property, the so-called monodromy nilpotent
cone is defined for each boundary point. We will find that, as a result of monodromy relations,
the monodromy nilpotent cones glue together to define a larger cone which can be identified
with the movable cone Mov(X) under mirror symmetry.
Studying birational geometry in mirror symmetry (or string theory) goes back to papers
by Morrison and his collaborators in the 90’s [2]. The birational geometry discussed in the
90’s was mostly for Calabi–Yau hypersurfaces in toric varieties, and it comes from the different
resolutions of ambient toric varieties. In this paper, we will study two specific examples of
complete intersection Calabi–Yau threefolds for which we have birational models in slightly
different form, and also have birational automorphisms of infinite order.
The construction of this paper is as follows: In Section 2, we will first recall some back-
ground material on mirror symmetry as formulated in the 90’s. Restricting our attentions to
three dimensional Calabi–Yau manifolds, we will summarize the basic properties of Calabi–Yau
manifolds called A- and B-structures. In Section 3, we will introduce a specific Calabi–Yau
This paper is a contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko
Yui. The full collection is available at http://www.emis.de/journals/SIGMA/modular-forms.html
mailto:hosono@math.gakushuin.ac.jp
mailto:hiromici@math.gakushuin.ac.jp
https://doi.org/10.3842/SIGMA.2018.039
http://www.emis.de/journals/SIGMA/modular-forms.html
2 S. Hosono and H. Takagi
threefold given by a complete intersection in P4 × P4, whose birational geometry and mirror
symmetry were studied in detail in previous works [26, 27]. We will describe its movable cone
by studying the geometry of birational models. In Section 4, we will report some results of
monodromy calculations, and describe the details of how the monodromy nilpotent cones glue
together by monodromy relations. In Section 5, we will present another complete intersection
given in P3×P3. Although there do not appear other birational models to this Calabi–Yau three-
fold than itself, we will observe interesting gluing property of monodromy nilpotent cones which
corresponds to the structure of the movable cone observed in [41]. Summary and discussions
will be presented in Section 6. There we will also describe the corresponding calculations for
a K3 surface in P3×P3 which has a parallel description to the complete intersection in P4×P4.
2 Classical mirror symmetry
2.1 Mirror symmetry of Calabi–Yau threefolds
Let us consider Calabi–Yau threefolds X and X∗ which will be taken to be mirror to each other.
For each of these, we have two different structures, called A-structure and B-structure.
2.1.1 A-structure of X
Let KX be the Kähler cone of X and κ1, . . . , κr ∈ H1,1(X,R) = H1,1(X) ∩ H2(X,R) be ge-
nerators of the Kähler cone, where for simplicity, we assume that the Kähler cone is a simplicial
cone in H2(X,R). Let κ be the Kähler class which corresponds to the polarization of X and
write κ by
κ = t1κ1 + · · ·+ trκr,
with ti > 0. The Lefschetz operator Lκ(−) := κ ∧ (−) defines a nilpotent linear action on the
even cohomology Heven(X) := ⊕pHp,p(X). In fact, this is a part of the Lefschetz sl(2,C) action,
and defines the following decomposition:
H0,0
H1,1
H2,2
H3,3
=
•
↓
• • •
↓ ↓ · · · ↓ Lκ.
• • •
↓
•
From the viewpoint of homological mirror symmetry, it is natural to replace Heven(X) with
the Grothendieck group K(X) (modulo torsion) which is an abelian group equipped the sym-
plectic form
χ(−,−) : K(X)×K(X)→ Z
with χ defined by χ(E ,F) :=
∑
(−1)i dimH i(X, E∗ ⊗ F) for vector bundles. Based on this
integral and symplectic structure on K(X), we can introduce the corresponding structure
on Heven(X,Q). A-structure of X is the nilpotent action Lκ on Heven(X,Q) with this inte-
gral and symplectic structure.
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 3
2.1.2 B-Structure of X∗
Let X∗ = X∗b0 and consider a smooth deformation family π : X∗ → B of X∗b0(b0 ∈ B) over
some open parameter space B. We denote by X∗b = π−1(b) the fiber over b ∈ B. Then we
have Kodaira–Spencer map ρb : TbB → H1(Xb, T X∗b ) which we assume to be an isomorphism.
Associated to this family, we naturally have the local system R3π∗CX∗ on B. In the 90’s, mirror
symmetry was recognized by finding some local family X|∆∗r → ∆∗r with special properties over
the product of punctured disc ∆∗r = (∆∗)r where ∆∗ = {z ∈ C | 0 < |z| < 1} and dimB = r.
The required properties for the local family are described by the monodromy representation
of the fundamental group π1(∆∗r) ' Zr for the local system R3π∗CX∗ restricted to over ∆∗r .
Let Ti represent the monodromy matrix corresponding to the i-th generator of π1(∆∗r) with
fixing a base point b0 ∈ ∆∗r . Assuming that all Ti are unipotent, we have nilpotent matrices
Ni = log Ti =
∑
k≥1
(−1)k−1
k (Ti − id)k. The set
Σ =
{∑
λiNi |λi ∈ R>0
}
(2.1)
is called monodromy nilpotent cone consisting of nilpotent matrices on H3(Xb0 ,Q). It is known
that each element of Σ defines the same monodromy weight filtration on H3(Xb0 ,Q) (see [16,
Theorem 1.9]). The following definition is due to Morrison [38].
Definition 2.1. The degeneration of the local family X|∆∗r → ∆∗r at the origin is called a large
complex structure limit (LCSL) if the following hold:
(1) All Ti, i = 1, . . . , r, are unipotent.
(2) Let Nλ =
∑
i λiNi λi > 0. This induces the monodromy weight filtration,
W0 = W1 ⊂ W2 = W3 ⊂ W4 = W5 ⊂ W6 = H3(Xb0 ,Q)
• ← • ← • ← •
• ← •
...
...
• ← •
(2.2)
with dimW0 = 1 and dimW2 = 1 + r.
(3) Let W0 = Qw0 and introduce a bi-linear form on W0 by 〈w0, w0〉 = 1. This defines mjk :=
〈w0, Njwk〉 for a Q-basis [w1], . . . , [wr] of W2/W0. Then the r × r matrix (mjk)1≤j,k≤r is
an invertible Q-matrix.
We note that there is a natural integral symplectic structure on H3(X∗b0 ,Z), and the mono-
dromy matrices Ti are given by integral and symplectic matrices if we fix a symplectic basis of
H3(X∗b0 ,Z). B-structure of X∗ at LCSL is defined to be such an integral and symplectic basis
of H3(X∗b0 ,Z) with the monodromy matrices Ti which are compatible with the filtration (2.2).
2.1.3 Mirror symmetry
In classical mirror symmetry, X is called a mirror to X∗ if the A-structure of X is isomorphic
to the B-structure of X∗, i.e., the two nilpotent actions Lκ and Nλ are identified together with
their integral and symplectic structures. To be more explicit, suppose we have a B-structure at
a LCSL. Since N4
λ = 0 and N3
λW6 ⊂W0, we have
NiNjNk = CijkN0
4 S. Hosono and H. Takagi
with a fixed rank one nilpotent matrix N0 satisfying NiN0 = 0. Corresponding to this products
of nilpotent matrices, we have, in the A-structure, the cup-product
κi ∪ κj ∪ κk = KijkV0, Kijk ∈ Z,
where V0 ∈ H3,3(X) normalized by
∫
X V0 = 1. After fixing a normalization of the matrix N0, we
have Cijk = Kijk if X and X∗ are mirror to each other, in particular we have Cijk ∈ Z≥0. In
fact, Cijk is the leading coefficient of the so-called Griffiths–Yukawa coupling, and Kijk is the
leading term of the quantum product. Mirror symmetry implies the equality between the two
in full orders under the so-called mirror map.
2.2 Birational geometry and mirror symmetry
Calabi–Yau threefolds often come with birational models. Mirror symmetry in such cases has
been studied in [39] and is known as topology change in physics [2]. The purpose of this paper
is to elaborate such cases in more details comparing the A-structure of X and the B-structure
of X∗. In the 90’s, Morrison considered the movable cone of X in the context of mirror symmetry
and also the topology change. We will push this perspective further by finding the corresponding
cone structure in terms of the monodromy nilpotent cones in the B-structure of X∗.
2.2.1 Movable cones of X
As above, let us assume that Calabi–Yau threefold X =: X1 comes with several other Calabi–
Yau threefolds Xi, i = 2, . . . , s, which are birational to each other. Let Ki ⊂ H2(Xi,R) be the
Kähler cone of Xi. Using the birational maps ϕi : X 99K Xi, these Kähler cones of Xi can be
transformed to the corresponding cones in H2(X,R). The convex hull of the union of these
cones is the movable cone Mov(X) of X. It is shown in [31] that the union of the transformed
Kähler cones defines a chamber structure to the movable cone Mov(X) (see also [39, Section 5]).
To work with the classical mirror symmetry, in fact, we have to consider the movable cone
in H2(X,R) ⊗ C using the complexified Kähler cones Ki +
√
−1H2(Xi,R). However, in this
paper, we will mostly focus on the structures in the real part of the complexified Kähler moduli.
2.2.2 Compactification of the moduli space Mcpx
X∗
Suppose that X∗ is mirror to X, i.e., we have a mirror family X∗ → B :=Mcpx
X∗ over a parameter
space Mcpx
X∗ on which we find a local (smooth) family X|∆∗r → ∆∗r ⊂ M
cpx
X∗ to describe the B-
structure which is mirror to the A-structure ofX. In the classical mirror symmetry of Calabi–Yau
complete intersections in toric varieties, there is a natural (toric) compactificationMcpx
X∗ [21, 25]
of the moduli space Mcpx
X∗ , and the geometry ∆∗r ⊂M
cpx
X∗ is characterized by the corresponding
normal crossing boundary divisors at the origin o ∈ ∆r = Cr.
The following properties can be observed for an abundance of examples of complete intersec-
tion Calabi–Yau manifolds:
Observation 2.2. Assume X and X∗ are Calabi–Yau threefolds which are mirror to each other.
If Calabi–Yau threefold X =: X1 has birational models Xi, i = 2, . . . , s, then there appear the
corresponding boundary points o =: o1 and oi, i = 2, . . . , s, given by normal crossing divisors
in Mcpx
X∗ such that
(1) oi are LCSLs, and
(2) the A-structures of Xi are isomorphic to the B-structures arising from oi.
Observation 2.3. Let X and X∗ be as above. Corresponding to the birational map ϕji : Xi 99K
Xj , there is a path connecting oi to oj and the connection matrix Mji of the B-structures such
that
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 5
(1) it preserves the monodromy weight filtrations, and
(2) it is integral and also compatible with the symplectic structures at each oi, i.e., tMjiΣjMji
= Σi for the symplectic matrices Σi representing the symplectic forms on H3(X∗boi
,Z).
Remark 2.4. Recently Calabi–Yau manifolds which are derived equivalent but not birational
each other have been attracting attention (see, e.g., [6, 27, 28, 35] and references therein).
These are called Fourier–Mukai partners after the original work by Mukai for K3 surfaces [40].
As shown in examples [6, 27, 46], if a Calabi–Yau threefold X has such Fourier–Mukai partners,
then corresponding boundary points exist in Mcpx
X∗ with the property (2) in Observation 2.3,
but losing the property (1). If we have both the property (1) and (2), then we can see that
the so-called prepotential for quantum cohomology is invariant up to quadratic terms under
analytic continuations (see Proposition 4.15 below), and hence the quantum cohomologies of
birational Calabi–Yau threefolds are essentially the same (see [37] for example). However, as
we see in [5, 23, 27, 43], quantum cohomologies of Fourier–Mukai partners are quite different to
each other.
In this paper we will focus on Calabi–Yau threefolds given by complete intersections in toric
varieties. Showing two examples which exhibit interesting birational geometry, we will make
Observation 2.3 more explicit, e.g., we will give precise descriptions about the path connecting
the boundary points. Also finding some monodromy relations, we will come to the following
observation:
Main result. Assume a Calabi–Yau threefold X and its mirror manifold X∗ have the properties
described in Observation 2.3. Then there are natural choices of path connecting oi and oj such
that the monodromy nilpotent cones defined for each oi inMcpx
X∗ are glued together. We identify
the resulting structure as the mirror counter part of the movable cone obtained by gluing Kähler
cones by birational maps.
The gluing will be achieved by finding monodromy relations coming from boundary divisors
which have multiple tangency with some component of the discriminant (see Section 4). When
writing the monodromy relations, we find a certain monodromy action of a distinguished form,
which we call “Picard–Lefschetz formula of flopping curves” based on the mirror correspondence
(cf. the same forms are known in physics literatures, [1, 9] for example, as strong coupling limits
associated to certain contractions of curves).
3 Complete intersection Calabi–Yau spaces
from Gorenstein cones
In this section, we describe mirror symmetry of a Calabi–Yau complete intersection of the form
X :=
(
P4|11111
P4|11111
)2,52
, (3.1)
i.e., a complete intersection of five general (1, 1) divisors in P4 × P4 which has Hodge numbers(
h1,1, h2,1
)
= (2, 52). In this section, we will study the A-structure of X.
3.1 Cones for complete intersections and Calabi–Yau manifolds
To describe the complete intersection X, let us note that we can write X = s−1(0) with a generic
choice of a section of the bundle O(−1,−1)⊕5 → P4 × P4. We describe this starting with the
affine cone over the generalized Segre embedding s1,1,1
(
P4 × P4 × P4
)
, which we write by
U0 := SpecC[λizjwk | 1 ≤ i, j, k ≤ 5]
6 S. Hosono and H. Takagi
with the homogeneous coordinates λi, zj , wk of P4’s. Let U → U0 be the blow-up of the cone
at the origin. It is easy to see that the exceptional divisor E is isomorphic to P4 × P4 × P4.
In fact, U is isomorphic to the total space of the line bundle O(−1,−1,−1) → P4 × P4 × P4.
Contracting one of the P4’s (m-th factor of P4
λ × P4
z × P4
w), we have three possible contractions
of U which fit in the following diagram:
U1
π1
yy
OO
��
U0 U2
π2oo
OO
��
U.
ee
oo
zz
U3
π3
ee
Again, it is easy to see that Uα → U0, α = 1, 2, 3, are small resolutions, and the geometries
of Uα are of the form O(−1,−1)⊕5 → P4×P4 that are birational to each other. It is worthwhile
noting that if we start with the cone over s1,1
(
P1 × P1
)
in the above construction, the resulting
geometry is the standard Atiyah flop for the small resolutions of the form O(−1)⊕O(−1)→ P1.
Definition 3.1. Consider the potential function on U0,
W =
∑
i,j,k
aijkλizjwk
with aijk ∈ C being chosen generically. Let Wα := π#
αW be the potential functions on Uα. We
denote the critical locus of Wα in each Uα by
Xα := Crit(Wα, Uα), α = 1, 2, 3.
Proposition 3.2. The critical locus Xα is a Calabi–Yau complete intersection of the form (3.1).
Proof. By symmetry, we only consider the case X1. To write the conditions for the criticality,
it is helpful to use the homogeneous coordinate for the small resolution U1, which is the total
space O(−1,−1)⊕5 → P4
z×P4
w. Let zi, wj denote the homogeneous coordinates of P4
z×P4
w and λi
be the fiber coordinate. Then the potential function is simply given by W1 =
∑
i,j,k aijkλizjwk,
which gives the conditions for the criticality ∂W1
∂λi
= ∂W1
∂zj
= ∂W1
∂wk
= 0. If we denote ∂W1
∂λi
=∑
j,k aijkzjwk =: fi(z, w), the conditions ∂W1
∂zj
= ∂W1
∂wk
= 0 may be arranged into a matrix form(
∇zf1 · · · ∇zf5
∇wf1 · · · ∇wf5
)( λ1
...
λ5
)
= 0.
The last equation gives the zero section {λ1 = · · · = λ5 = 0} ' P4
z × P4
w and the conditions
f1(z, w) = · · · = f5(z, w) = 0 give a smooth complete intersection in the zero section if we
choose aijk sufficiently general. �
Proposition 3.3 ([26, 27]). Xα and Xβ, α 6= β, are birational. The birational maps ϕβα : Xα 99K
Xβ are given by the Atiyah flops associated to the contractions of 50 P1s, which we summarize
in the following diagram:
X1 X2 X3 X1,
Z2 Z3 Z1
π21 �� π22�� π32 �� π33�� π13 �� π11��
oo // oo // oo //
(3.2)
where Z1 ⊂ P4
z, Z2 ⊂ P4
w and Z3 ⊂ P4
λ are determinantal quintics defined by the 5× 5 matrices(∑
zjaijk
)
,
(∑
wkaijk
)
and
(∑
λiaijk
)
, respectively.
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 7
We refer the references [26, 27] for the proof of the above proposition.
Remark 3.4. In the above proposition, we naturally come to birational Calabi–Yau complete
intersections. Some remarks related to this are in order:
1. Uα’s are birational to each other since they are all toric varieties with the same algebraic
torus contained as a dense subset. In fact, they all have the form O(−1,−1)⊕5 → P4×P4.
However, when defining Xα as the critical locus of the potential function, the zero section
of O(−1,−1)⊕5 → P4 × P4 is specified by the criticality condition. Hence, that Uα’s are
birational does not imply that Xα’s are birational. The fact that Xα’s are birational comes
from different reasons as described in the above proposition.
2. The affine cone construction here is an example of more general method in toric geom-
etry due to Batyrev and Borisov [4]. There, the affine cone is replaced by the so-called
Gorenstein cones, and actually a pair of reflexive Gorenstein cone (C∇, C∆) to describe
mirror symmetry. The birational geometry we observed in the above proposition has been
described by the property of so-called nef-partitions of ∇ by Batyrev and Nil [3]. They
have found that the two different (but isomorphic) nef-partitions
∇ = ∇1 +∇2 + · · ·+∇s = ∇′1 +∇′2 + · · ·+∇′s
sometimes results in dual nef-partitions
∆ = ∆1 + ∆2 + · · ·+ ∆s, ∆′ = ∆′1 + ∆′2 + · · ·+ ∆′k
with ∆ and ∆′ having completely different shapes to each other. We can describe our
birational Calabi–Yau threefolds in this general setting. See references [7, 12] for recent
works which shed light on this general phenomenon from the derived categories of Calabi–
Yau threefolds.
3.2 Movable cone of X := X1
Let us note that the Kähler cone of X(= X1) is given by KX = R>0H1 + R>0H2 with the pull-
backs H1 = π∗11HZ1 and H2 = π∗21HZ2 of the hyperplane classes HZi of Zi, where πji : Xi → Zj
is the projection in the diagram (3.2).
Lemma 3.5.
(1) Let KX2 = R>0LZ2 + R>0LZ3 be the Kähler cone with the generators LZ2 = π∗22HZ2 and
LZ3 = π∗32HZ3. By the birational map ϕ21 : X1 99K X2, the Kähler cone is transformed to
ϕ∗21(KX2) = R>0H2 + R>0(4H2 −H1).
(2) Similarly, let KX3 = R>0MZ3 + R>0MZ1 be the Kähler cone of X3 generated by MZ3 =
π∗33HZ3 and MZ1 = π∗13HZ1, then we have
ϕ∗31(KX3) = R>0(4H1 −H2) + R>0H1
for the birational map ϕ31 : X1 99K X3.
Proof. See Appendix A. �
Lemma 3.6. With the divisors LZ2, LZ3 and MZ3, MZ1 defined as above, we have
ϕ∗32(MZ1) = 4LZ3 − LZ2 , ϕ∗32(MZ3) = LZ3
for the birational map ϕ32 : X2 99K X3.
8 S. Hosono and H. Takagi
Proof. The second relation holds by definition. For the first relation, see Appendix A. �
Now, we define the following composite of the birational maps:
ρ := ϕ13 ◦ ϕ32 ◦ ϕ21
with the convention ϕij = ϕ−1
ji : Xj 99K Xi (see the diagram (3.2)).
Lemma 3.7. The birational map ρ is not an automorphism of X. It is of infinite order.
Proof. We show that
ρ∗H1 = −4H1 + 15H2, ρ∗H2 = −15H1 + 56H2 (3.3)
for ρ∗ = ϕ∗21 ◦ ϕ∗32 ◦ ϕ∗13. Since ϕ∗13 =
(
ϕ−1
31
)∗
= (ϕ31)∗ and using the relations ϕ∗31(MZ3) =
4H1 −H2, ϕ
∗
31(MZ1) = H1 in Lemma 3.5(2), we have
MZ3 = 4MZ1 − ϕ∗13(H2), MZ1 = ϕ∗13(H1).
Then, using Lemmas 3.5 and 3.6, it is straightforward to evaluate ρ∗(Hi), e.g., ρ∗(H1) = ϕ∗21 ◦
ϕ∗32(MZ1) = ϕ∗21(4LZ3 − LZ2) = 4(4H2 − H1) − H2. From these actions of ρ∗, we see that
ρ∗(KX) 6= KX and hence ρ /∈ Aut(X). Also, expressing the linear action (3.3) by a matrix(−4 −15
15 56
)
, we see that ρ has an infinite order. �
Proposition 3.8. Suppose Xi 6' Xj, i 6= j, then the groups of birational maps of Xi are given
by
Bir(Xi) = Aut(Xi) · 〈ϕi1 ◦ ρ ◦ ϕ1i〉.
Proof. Since arguments are similar to [41, Lemma 6.4], here we only give a rough sketch. Also,
we only describe the case i = 1, ϕ11 = idX . Take a birational map τ : X 99K X. We denote
by E(τ) the locus where τ is not defined or non-isomorphic. Consider an ample divisor D and
its transform D′ = (τ−1)∗D. Under this setting, we consider the two cases: (i) If D′ is nef, then
using [33, Lemma 4.4] we have D|E(τ−1) ≡ 0, i.e., numerically equivalent to zero. Since D is
ample, this implies E
(
τ−1
)
= ∅, i.e., τ ∈ Aut(X). (ii) If D′ is not nef, the restriction D′|E(τ)
is not nef, too. This is because if D′|E(τ) were nef, then D′ = (τ−1)∗D must be nef because D
is ample. Therefore D′|E(τ) is not nef and there exists a curve C ⊂ E(τ) such that D′ · C < 0.
Now, since KX |E(τ) ≡ 0, we know that KX + εD′, 0 < ε � 1, is not nef and (X, εD′) is klt.
From the theory of minimal models, we know that there exists an extremal ray of NE(X) and
its associated contraction, which must be either X → Z1 or X → Z2 up to automorphisms.
Now, corresponding to these two possibilities, we make the following diagrams:
X X2 X
Z1
X X3 X.
Z2
or
τ
))ϕ21 //
�� ��
τ
))ϕ31 //
�� ��
(3.4)
Depending on the two cases, we set D′′ =
(
ϕ21τ
−1
)
∗D or D′′ =
(
ϕ31τ
−1
)
∗D and consider
inductively the above two cases (i) and (ii) again. Due to [33, Theorem 3.5], this process
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 9
terminates arriving at the case (i) in the end. We can deduce that there are only two possibilities
under the assumption X 6' Xi, i = 2, 3:
X = X1
ϕ21
99K X2
ϕ32
99K X3
ϕ13
99K X1
ϕ21
99K X2 · · ·
ϕ13
99K X1
ϕL−−→
∼
X1 = X,
X = X1
ϕ31
99K X3
ϕ23
99K X2
ϕ12
99K X1
ϕ31
99K X3 · · ·
ϕ12
99K X1
ϕR−−→
∼
X1 = X.
ρ 55
ρ−1 55
Corresponding to these two, we have the decomposition τ = ϕLρ
n or τ = ϕR(ρ−1)m with
ϕL,R ∈ Aut(X). �
Remark 3.9. We use the assumption Xi 6' Xj at the very end of the above proof. If X1 ' Xi,
then it is easy to deduce that we only have to include ϕi1 in the generators of Bir(X1). Similar
modification in Bir(Xi) is required if Xi ' Xj , i 6= j. These do not affect the form of the
movable cone determined below. The assumption in the above proposition has been made just
for simplicity.
Let us denote by Mov(Xi) be the movable cones generated by movable divisors on Xi. Since
the transforms of movable divisors by flops are movable, we have
Mov(X) = Mov(X1) = ϕ∗21 Mov(X2) = ϕ31 Mov(X3).
The following result is known by [14, Lemma 1]. For completeness of our arguments, we present
it here with a general proof.
Proposition 3.10. The closure of the movable cone Mov(X) is given by
Mov(X) = R≥0
(
−H1 + (2 +
√
3)H2
)
+ R≥0
(
H1 + (−2 +
√
3)H2
)
. (3.5)
Proof. By Lemmas 3.5 and 3.6, it is easy to see that the closure of the set ϕ∗21(KX2) ∪ KX1 ∪
ϕ∗31(KX3) is given by
C123 := R≥0(4H2 −H1) + R≥0(4H1 −H2).
We define
M :=
⋃
n∈Z
(ρ∗)nC123 := 〈ρ∗〉 · C123.
Then, from a linear algebra, it is straightforward to see that the r.h.s. of (3.5) coincides with the
closure M . Since any automorphism of Xi preserves the generators of KXi or exchanges them,
using Proposition 3.8, we have ∪iϕ∗i1(Bir(Xi)
∗KXi) = M . Hence we have M ⊂ Mov(X).
To show the other inclusion, take a rational point d ∈ Mov(X). There exist m � 1 and an
effective movable divisor D such that md = [D]. If D is nef, then d ∈ KX and hence d ∈ M .
If D is not nef, we do the same inductive process as in the proof of Proposition 3.8 and find
a birational map τ : X 99K Xi, τ ∈ 〈ρ, ϕ21, ϕ31〉, such that D′ = τ∗D is a nef divisor on Xi, i.e.,
D′ ∈ KXi . Namely, we have D = τ∗D′ ∈ KX and KX ⊂ M , which imply Mov(Xi)(Q) ⊂ M .
Hence we have Mov(X) ⊂M . �
10 S. Hosono and H. Takagi
Figure 1. Movable cone Mov(X) in H2(X,R). The rays accumulate to the boundary rays of slopes
−2−
√
3 and −2 +
√
3.
3.3 Mirror symmetry of X
For the complete intersection Calabi–Yau threefoldsX (= X1), the mirror family can be obtained
by a straightforward application of the Batyrev–Borisov toric mirror construction. However, the
construction involves complications in combinatorics for toric geometry. In our case, we can
avoid these complications and find the mirror family of X by the so-called orbifold mirror
construction starting with a special family [26].
Define the following special family of X1:
Xsp := {ziwi + aziwi+1 + bzi+1wi = 0, i = 1, . . . , 5} ⊂ P4
z × P4
w,
where the indices of zi, wj should be considered modulo 5.
Proposition 3.11. For general values of a, b, we have the following properties:
(1) Xsp is singular along 20 lines of singularity of A1 type.
(2) There exists a crepant resolution X∗ → Xsp with X∗ being a Calabi–Yau threefold with
h1,1(X∗) = 52, h2,1(X∗) = 2.
(3) The resolution X∗ parametrized by (a, b) ∈ C2 defines a family X∗ → Mcpx
X∗ \ Dis with
Mcpx
X∗ = P2 and Dis = D1 ∪ D2 ∪ D3 ∪ Dis0 where Di are the coordinate lines of P2
and Dis0 is an irreducible (singular) curve of degree 5. The fiber over
[
a5, b5, 1
]
6∈ Dis is
given by the resolution X∗ with (a, b).
Proof. Proofs of these properties are given in [26, Theorems 5.11 and 5.17]. �
We can verify that all the properties in Observation 2.2 hold for the family X∗ → P2 \Dis.
Proposition 3.12. We set
o1 = D1 ∩D2 = [0, 0, 1], o2 = D2 ∩D3 = [1, 0, 0], o3 = D3 ∩D1 = [0, 1, 0].
All these boundary points o1, o2, o3 are LCSLs whose B-structures are identified with the A-
structures of the birational models X1, X2 and X3, respectively. See Fig. 2 in the next section.
The above proposition has been derived by introducing integral and symplectic structures at
each oi and calculating the monodromies around the divisors Di, see [26, Section 6.3] for details.
Our focus in what follows will be gluing the monodromy cones (2.1) which are defined for each
boundary point oi.
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 11
4 Gluing monodromy nilpotent cones I
For the example in the preceding section, we will find a path oi → oj which we can identify
with the birational map ϕji : Xi 99K Xj as described in Observation 2.3. We will find that
the monodromy nilpotent cones (2.1) at each boundary point are naturally glued together by
the monodromy relations coming from the path. Also, in the next section (Section 5) we will
study another interesting example, which has no other birational models other than itself but
has a birational automorphism of infinite order.
4.1 B-structures of X∗
Associated to the family π : X∗ →Mcpx
X∗ \ Dis, we have the local system R3π∗CX∗ which intro-
duces the Gauss–Manin system on the moduli space, or equivalently the Picard–Fuchs differential
equation for the period integrals of holomorphic three form. This Picard–Fuchs equation has
been studied in our previous work [26], where we have described the B-structure for the bound-
ary points oi, i.e., the integral and symplectic basis for the local solutions as well as integral
monodromy matrices using the central charge formula given in [19, 20] which goes back to the
study of GKZ system [15] in the 90’s (see [22, 25] for details). Here we briefly recall the integral
and symplectic basis referring to [26] for its explicit form, and define the monodromy nilpotent
cones for each oj from the monodromy matrices calculated there.
4.1.1 B-structure at o1
Let [−x,−y, 1] ∈ P2 be the affine coordinate with the origin o1 (where the minus signs are
required to have the canonical integral and symplectic structure based on the central charge
formula). The canonical, integral and symplectic structure appears from a unique power series
solution w0(x, y) of the Picard–Fuchs differential equation around the origin o1. Including the
logarithmic solutions, the result can be arranged as follows:
Π(x, y) =
t(
w0(x, y), w
(1)
1 (x, y), w
(1)
2 (x, y), w
(2)
2 (x, y), w
(2)
1 (x, y), w(3)(x, y)
)
=
t
(∫
A0
Ωx,
∫
A1
Ωx,
∫
A2
Ωx,
∫
B2
Ωx,
∫
B1
Ωx,
∫
B0
Ωx
)
, (4.1)
where {A0, A1, A2, B2, B1, B0} ⊂ H3(X∗bo ,Z) is a symplectic basis satisfying Ai ∩ Bj = δij ,
Ai ∩ Aj = Bi ∩ Bj = 0 representing the integral and symplectic solutions of the Picard–Fuchs
equation [26, Section 6.3.1]. The monodromy matrix Tx of Π(x, y) for a small loop around x = 0
and similarly Ty for y = 0 have been determined as follows:
Tx =
1 0 0 0 0 0
1 1 0 0 0 0
0 0 1 0 0 0
5 10 10 1 0 0
2 5 10 0 1 0
−5 −3 −5 0 −1 1
, Ty =
1 0 0 0 0 0
1 1 0 0 0 0
1 0 1 0 0 0
2 10 5 1 0 0
5 10 10 0 1 0
−5 −5 −3 −1 0 1
.
We define
B1 := {α0, α1, α2, β2, β1, β0} ⊂ H3(X∗bo ,Z) (4.2)
to be the dual basis satisfying
∫
Ai
αj = δ ji =
∫
Bi
βj and
∫
Ai
βj =
∫
Bi
αj = 0. Since the
monodromy actions on the period integrals, i.e., on H3(X∗bo ,Z), are translated into the dual
space via the transpose and inverse, we define the linear action Nλ =
∑
λiNi on H3(X∗bo ,Z) by
N1 := − t(log Tx), N2 := − t(log Ty).
12 S. Hosono and H. Takagi
Then we define the monodromy nilpotent cone at o1 by
Σo1 :=
{∑
λiNi |λi > 0
}
⊂ End
(
H3(Xb0 ,Q)
)
. (4.3)
For general values of λi > 0, it is easy to see that the nilpotent matrixNλ induces the monodromy
weight filtration W0 ⊂W2 ⊂W4 ⊂W6 = H3(X∗bo ,Q) given by
W0 = 〈α0〉, W2 = 〈α0, α1, α2〉,
W4 = 〈α0, α1, α2, β2, β1〉, W6 = 〈α0, α1, α2, β2, β1, β0〉. (4.4)
Using the matrices N1, N2, it is easy to see the following property:
Proposition 4.1. We have
NiNjNk = CijkN0
with totally symmetric Cijk given by C111 = C222 = 5, C112 = C122 = 10 and Cijk = 0 for other
cases, and N0 =
(
0 1
O5 0
)
where O5 is the zero matrix of size 5× 5.
Remark 4.2. As we see above, the monodromy matrices of the period integrals act onH3(X∗bo ,Z)
while the monodromy weight filtration is defined in the dual space H3(X∗bo ,Z). Hence, we
translate any monodromy matrix A obtained from the analytic continuations of the period integ-
ral Π(x, y) to the corresponding matrix A in the dual space by A = tA−1.
4.1.2 B-structures at o2, o3
In a similar way to the last paragraph, we determine the B-structure from the boundary
points o2 and o3, which are given by the origins of the affine charts [1,−y′,−x′] ∈ P2 and
[−x′′, 1,−y′′] ∈ P2. As described in detail in [26, Section 6.3.1], we have the canonical integral
and symplectic basis
Π′(x′, y′) = x′Π(x′, y′) and Π′′(x′′, y′′) = y′′Π(x′′, y′′) (4.5)
in terms of the same Π(x, y) as (4.1) for o2 and o3, respectively. Since both of (4.5) have
essentially the same form as Π(x, y), we have
T′x′ = T′′x′′ = Tx and T′y′ = T′′y′′ = Ty (4.6)
for the monodromy matrices with the base points b′o and b′′o near the origins. Hence for o2 and o3
we have isomorphic B-structures with
Ñ ′1 = log T ′x′ , Ñ ′2 = log T ′y′ and Ñ ′′1 = log T ′′x′′ , Ñ ′′2 = log T ′′y′′ ,
where T ′x′ = ( tT′x′)
−1, T ′y′ = ( tT′y′)
−1 and similarly for T ′′x′′ , T
′′
y′′ . These nilpotent matrices
determine the respective monodromy weight filtrations in H3(X∗b′o ,Q) and H3(X∗b′′o ,Q) with the
basis {
α′0, α
′
1, α
′
2, β
′
2, β
′
1, β
′
0
}
and
{
α′′0, α
′′
1, α
′′
2, β
′′
2 , β
′′
1 , β
′′
0
}
, (4.7)
as described above. We denote the monodromy nilpotent cones at o2 and o3 by
Σ′o2
=
{∑
λiÑ
′
i |λi > 0
}
⊂ End
(
H3(X∗b′o ,Q)
)
,
Σ′′o3
=
{∑
λiÑ
′′
i |λi > 0
}
⊂ End
(
H3(X∗b′′o ,Q)
)
. (4.8)
These are the B-structures which we identify with the A-structures of the birational models X2
and X3, respectively, in [26].
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 13
4.2 Gluing the monodromy nilpotent cones
The monodromy matrices are transformed by conjugation when the base point is changed along
a path. We can transform the monodromy nilpotent cones (4.8) into H3(X∗bo ,Q) once we fix
paths pb′0←bo and pb′′0←bo . Let us denote by ϕb′obo the resulting isomorphism ϕb′obo : H3(X∗bo ,Q) '
H3(X∗b′o ,Q) and similarly for ϕb′′o bo . We define the transforms of the nilpotent cones (4.8) by
these isomorphisms by
Σo2 := (ϕb′obo)−1Σ′o2
ϕb′ob0 , Σo3 := (ϕb′′o bo)−1Σ′′o3
ϕb′′o bo .
Then the cones Σo2 and Σo3 are generated by
N ′i := (ϕb′obo)−1Ñ ′iϕb′ob0 , N ′′i := (ϕb′′o bo)−1Ñ ′′i ϕb′′o b0 , i = 1, 2,
respectively. Note that Σo1 , Σo2 , Σo3 are cones in End
(
H3(X∗bo ,Q)
)
.
4.2.1 Path po2←o1
The transform Σo2 of the nilpotent cone obviously depends on the choice of the path. Looking
the moduli space Mcpx
X∗ closely, we find that there is a natural choice of the path by which the
cone Σo2 is glued with Σo1 along a common face (boundary ray) of them.
The moduli space Mcpx
X∗ has been studied in detail in [26]. Here we recall the structure of
the discriminant Dis = Dis0 ∪ Dx ∪ Dy ∪ Dz. As we schematically reproduce the results in
Fig. 1, the irreducible component Dis0 of the discriminant touches the divisor Dy = {y = 0} at
(x, y) = (1, 0) with fifth-order tangency as we can see in the expression
Dis0 =
{
(1− x− y)5 − 54xy(1− x− y)2 + 55xy(xy − x− y) = 0
}
.
We introduce the affine chart C2
(1,0) with the origin (1, 0). After blowing-up at the origin five
times, we can remove the tangential intersection of the proper transform D̃is0 of Dis0 with the
exceptional divisors (see Fig. 2). We denote the exceptional divisors by E1, . . . , E5.
Definition 4.3. Let q12 be a point near the intersection E1 ∩Dy, and bo, b
′
o be points near the
origins o1 and o2, respectively. We define a path pb′o←b0 to be the composite path pb′o←q12◦pq12←bo
of the following straight lines:
pq12←bo = {(1− t)bo + tq12 | 0 ≤ t ≤ 1},
pb′o←q12 = {(1− t)q12 + tb′o | 0 ≤ t ≤ 1}.
4.2.2 The isomorphisms ϕb′ob0
, ϕb′′o b
′
o
and ϕbob′′o
We first calculate the connection matrix of the local solution Π(x, y) along the path pb′o←bo .
Proposition 4.4. With respect to the basis (4.2) and (4.7), the isomorphism ϕb′obo : H3(X∗bo ,Q) '
H3(X∗b′o ,Q) along the path pb′o←bo is given by
ϕb′obo =
−1 0 0 0 0 0
0 1 −4 2 25 0
0 0 −1 0 −2 0
0 0 0 −1 −4 0
0 0 0 0 1 0
0 0 0 0 0 −1
.
This isomorphism preserves the monodromy weight filtrations and also the symplectic structures
described in Section 4.1.2.
14 S. Hosono and H. Takagi
Figure 2. Blowing-up the moduli space Mcpx
X∗ = P2. To remove the tangential intersections at [1, 1, 0],
[1, 0, 1], [0, 1, 1], we blow-up five times at each of the three points. The exceptional divisors E1, E′1,
E′′1 are normal crossing with the proper transform D̃is0 of the discriminant. The affine coordinates are
introduced by the relations [−x,−y, 1] = [1,−y′,−x′] = [−x′′, 1,−y′′].
Proof. To determine the matrix form of ϕb′obo , we do first the analytic continuation of the
period integral Π(x, y) along the path pq12←bo by making local solutions around q12 = E1 ∩Dy
in terms of the blow-up coordinates s1 = x− 1, s2 = y
(1−x)5 which represent q12 by s1 = s2 = 0.
There are two local solutions which are given by regular powerseries, and others contain log-
arithmic singularities given by log s1 and log s2, . . . , (log s2)3. For a fixed value of y, |y| � 1,
we analytically continue these solutions to Π(x, y) as functions of s1 = x − 1. Note that, un-
der the analytic continuation, the powers of log y are unchanged. Hence the connection matrix
follows from the analytic continuation of the period integrals Π(x, 0) where we set log y = 0
and y = 0. In our actural calculation, we set s2 = 0 and log s2 = −5 log(1 − x) for the lo-
cal solutions around (s1, s2) = (0, 0), and relate these solutions numerically to Π(x, 0) using
powerseries expansions with sufficiently high degrees. In a similar way, we can calculate the
connection matrix for the latter half pb′o←q12 of the path pb′o←bo . Actually, we can avoid the
above numerical calculations finding an analytic formula for Π(x, 0). However, since the details
are technical, we will report them elsewhere. It is clear that the connection matrix ϕb′obo pre-
serves the filtrations since it is block diagonal with respect to the basis compatible with the
filtrations W0 ⊂ W2 ⊂ W4 ⊂ W6 = H3(X∗bo ,Q) and W ′0 ⊂ W ′2 ⊂ W ′4 ⊂ W ′6 = H3(X∗b′o ,Q).
Moreover, we can verify directly that it preserves the symplectic structure given by (4.7). �
From the forms of period integrals given in (4.5), it is easy to deduce that we have the
isomorphisms
ϕb′′o b′o : H3(X∗b′o ,Z) ' H3(X∗b′′o ,Z) and ϕbob′′o : H3(X∗b′′o ,Z) ' H3(X∗bo ,Z)
by simply exchanging the bases α1 ↔ α2 and β1 ↔ β2 suitably, i.e., ϕb′′o b′o = ϕb′obop23p45
and ϕbob′′o = p23p45ϕb′obop23p45 with the permutation matrices pij for the transposition (i, j).
Explicitly, they are given by
ϕb′′o b′o =
−1 0 0 0 0 0
0 −4 1 25 2 0
0 −1 0 −2 0 0
0 0 0 −4 −1 0
0 0 0 1 0 0
0 0 0 0 0 −1
, ϕbob′′o =
−1 0 0 0 0 0
0 −1 0 −2 0 0
0 −4 1 −25 2 0
0 0 0 1 0 0
0 0 0 −4 −1 0
0 0 0 0 0 −1
.
Here we note that these isomorphisms preserve the monodromy weight filtrations and also the
symplectic structures described in Sections 4.1.1 and 4.1.2. Also it should be noted that we have
verified Observation 2.3 in Section 2.2.2 in the present case.
Let us introduce the following notation:
ϕ̌21 := ϕb′obo , ϕ̌32 := ϕb′′o b′o , ϕ̌13 := ϕbob′′o
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 15
and also set ϕ̌ij := ϕ̌−1
ji . As this notation indicates, we expect certain correspondence of these ϕ̌ij
to the birational maps ϕij : Xj 99K Xi under mirror symmetry. In order to make this more
explicit, we note the groupoid structure associated to the isomorphisms ϕ̌ij .
Definition 4.5. We denote by G{1,2,3} the groupoid generated by ϕ̌21, ϕ̌32, ϕ̌13.
Let Gij be the subset of G{1,2,3} consisting of elements ϕ̌ii1ϕ̌i1i2 · · · ϕ̌ikj , k ≥ 0. It is easy to
see that
G11 =
{
ρ̌n |n ∈ Z
}
, G21 =
{
ϕ̌21ρ̌
n |n ∈ Z
}
, G31 =
{
ϕ̌31ρ̌
n |n ∈ Z
}
,
where set ρ̌ := ϕ̌13ϕ̌32ϕ̌21.
4.2.3 Groupoid actions on the nilpotent cones
We define the following conjugates of the nilpotent cones (4.3) and (4.8):
Σ(n)
o1
:=
(
ρ̌−1
)n
Σo1 ρ̌
n,
Σ(n)
o2
:=
(
ρ̌−1
)n
ϕ̌−1
21 Σ′o2
ϕ̌21ρ̌
n =
(
ρ̌−1
)n
Σo2 ρ̌
n,
Σ(n)
o3
:=
(
ρ̌−1
)n
ϕ̌−1
31 Σ′′o3
ϕ̌31ρ̌
n =
(
ρ̌−1
)n
Σo3 ρ̌
n.
These are cones in End
(
H3(X∗bo ,R)
)
and generalize the nilpotent cones Σok = Σ
(0)
σk , k = 1, 2, 3,
introduced in the beginning of this subsection. It is easy to see that these cones are generated
by
Ni(n) :=
(
ρ̌−1
)n
Niρ̌
n, N ′i(n) :=
(
ρ̌−1
)n
N ′i ρ̌
n, N ′′i (n) :=
(
ρ̌−1
)n
N ′′i ρ̌
n,
respectively, where we set N ′i := ϕ̌−1
21 Ñ
′
i ϕ̌21 and N ′′i := ϕ̌−1
31 Ñ
′′
i ϕ̌31, i = 1, 2.
4.2.4 Monodromy relations
To see how the (closure of the) cone Σo2 = Σ
(0)
o2 is connected to (that of) Σo1 = Σ
(0)
o1 , we calculate
the generators N ′i in End
(
H3(X∗bo ,Z)
)
. By the definition of N ′i , it suffices to calculate
Tx′ := ϕ̌−1
21 T
′
x′ϕ̌21, Ty′ := ϕ̌−1
21 T
′
y′ϕ̌21,
since we can use T ′x′ = Tx, T ′y′ = Ty for the local monodromy matrices as we remarked
in (4.6). Similarly, using the connection matrix along the path pq12←bo , we can express the
local monodromy around the exceptional divisor E1 as a linear (integral and symplectic) action
on H3(X∗bo ,Z) which we denote by a matrix TE1 using the basis B1 in (4.2).
Proposition 4.6 (‘Picard–Lefschetz formula’ for flopping curves). Using the basis B1 in (4.2),
we have
TE1 =
1
1 50
1
1
1
1
, i.e.,
α1 → α1 + 50β1,
β1 → β1,
αi = αi, βi = βi, i 6= 1.
Proof. As sketched briefly in the proof of Proposition 4.4, we make the local solutions of
the Picard–Fuchs equation around the point of the blow-up q12 = E1 ∩ Dy, and calculate the
local monodromy around the divisor E1. The claimed monodromy follows from the analytic
continuation of the local solutions in the period integral Π(x, y) near the origin o1. In our actual
calculations, we only have powerseries expressions for the local solutions around q12 and evaluate
them numerically for the analytic continuation. However, as in Proposition 4.4, we can attain
sufficient precision having an analytic formula for Π(x, 0). �
16 S. Hosono and H. Takagi
Remark 4.7. The ‘Picard–Lefschetz formula’ above is written using the symplectic basis
{αi, βj} of H3(X∗bo ,Z). When we translate this into the dual basis {Ai, Bj} of H3(X∗bo ,Z),
we have
A1 → A1, B1 → B1 − 50A1
with the rest of the basis left invariant. This should be contrasted to the genuine Picard–
Lefschetz monodromy
A0 → A0 +B0, B0 → B0,
which we can see for the monodromy transformation around the proper transform D̃is0 of the
discriminant. In the latter case, we see the topology of the cycles as A0 ≈ T 3, B0 ≈ S3, where S3
is a vanishing cycle and T 3 is its dual torus cycle. Recently, the construction of the Ak-cycles
(k 6= 0) has been discussed in general in [44]. It is interesting to see how the dual Bk-cycles
are constructed, and how the above ‘Picard–Lefschetz formula’ are explained by the geometry
of these cycles.
Proposition 4.8. We have the following monodromy relations:
Tx′ = T−1
E1
T−1
x T 4
y , Ty′ = Ty. (4.9)
Proof. Recall that we have the relations Tx = ( tTx)−1, Ty = ( tTy)
−1 (see Remark 4.2). Then
both the relations can be verified directly using the explicit forms of Tx, Ty given in Sec-
tion 4.1.1 and Tx′ , Ty′ , TE1 above. The second relation also follows from the fact that the divisor
{y = 0} = {y′ = 0} intersects normally with the exceptional divisor E1 of the blowing-up. �
We have arrived at (4.9) by explicit monodromy calculations. It is natural to expect to have
a conceptual derivation of (4.9) by studying mirror symmetry of conifold transitions, but we
have to this to future investigations. Instead, in the rest of this section, we will interpret the
monodromy relation (4.9).
Proposition 4.9. The following properties hold:
(1) Generators N ′i are expressed as
N ′1 = 4N2 −N1 + ∆′1,0, N ′2 = N2,
where ∆′1,0 is a non-zero element of End
(
H3(X∗bo ,R)
)
which annihilates the subspace W2,
i.e., ∆′1,0|W2 = 0.
(2) The monodromy nilpotent cones Σo2 = R>0N
′
1 + R>0N
′
2 and Σo1 glue together along
N ′2 = N2. They are not in a two dimensional plane in End
(
H3(X∗bo ,R)
)
.
Proof. The properties in (1) are based on explicit calculations using (4.9). The second relation
N ′2 = log Ty′ = log Ty = N2 is clear. For the first relation, by evaluating the matrix logarithms,
we have
∆′1,0 = N ′1 − (4N2 −N1) =
0 0 0 0 25 − 25
3
0 0 0 0 −50 25
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
.
From this triangular form, we see the claimed property of ∆′1,0 (see also (4.4)). The claims in (2)
are clear from (1) and also from the fact that the cone Σo1 is generated by N1 = log Tx and
N2 = log Ty. �
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 17
Remark 4.10. (1) It should be observed that, under the identification
LZ3 ↔ N ′1, LZ2 ↔ N ′2 and H1 ↔ N1, H2 ↔ N2,
Proposition 4.9 above is the mirror counter part for the gluing of Kähler cones described in
Lemma 3.5.
(2) If the first monodromy relation of (4.9) were Tx′ = T−1
x T 4
y , then we would have
N ′1 = 4N2 −N1, N ′2 = N2,
since Tx and Ty are commutative. These relations are exactly the same as those we have seen
in Lemma 3.5. However the presence of TE1 prevents this exact correspondence. We will see
that TE1 represents the first order quantum correction coming from the 50 flopping curves of the
contraction X1 99K Z1. Thus the gluing relation found in Proposition 4.9(1) naturally encodes
the first order quantum corrections.
4.2.5 Gluing nilpotent cones
Before going into general descriptions, it will be helpful to see that the cone Σo1 is glued with Σo3
along N1 in a similar way as above. Let us define
Tx′′ := ϕ̌−1
31 T
′′
x′′ϕ̌31, Ty′′ := ϕ̌−1
31 T
′′
y′′ϕ̌31,
and also TE′′1 for the monodromy matrix around the exceptional divisor E′′1 . Observing the
symmetry in Fig. 2 and (4.5), it is easy to deduce the following monodromy relations
Tx′′ = Tx, Ty′′ = T−1
E′′1
T−1
y T 4
x (4.10)
with TE′′1 = p23p45TE1p23p45 in End
(
H3(X∗bo ,Z)
)
, where pij are the permutation matrices. Since
the generators of the cone Σo3 are given by N ′′1 = log Ty′′ and N ′′2 = log Tx′′ , we can evaluate
these as
N ′′1 = N1, N ′′2 = 4N1 −N2 + ∆′′2,0, (4.11)
where ∆′′2,0 is given by ∆′′2,0 = p23p45∆′1,0p23p45 with the vanishing property ∆′′2,0|W2 = 0. As
before, ∆′′2,0 is a non-vanishing element. Hence, the nilpotent cones Σo1 and Σo3 glue together
along the common half line R≥0N1 but do not lie on the same plane. Now we generalize these
properties in the following proposition.
Proposition 4.11.
(1) The matrix ρ̌ preserves the monodromy weight filtration
W0 ⊂W2 ⊂W4 ⊂W6 = H3(X∗bo ,Q).
(2) The (closures of the) monodromy nilpotent cones Σ
(n)
o1 , Σ
(n)
o2 , Σ
(n)
o3 glue sequentially as in
. . . ,Σ(1)
o2
, Σ(1)
o1
, Σ(1)
o3
, Σo2 , Σo1 , Σo3 , Σ(−1)
o2
, Σ(−1)
o1
, Σ(−1)
o3
, . . . .
(3) The generators of the cones satisfy
(i) (N1(n), N2(n)) = (N1, N2)
(
0 −1
1 4
)3n
+ (∆1,n,∆2,n),
18 S. Hosono and H. Takagi
(ii) (N ′1(n), N ′2(n)) = (N ′1, N
′
2)
(
0 −1
1 4
)−3n
+ (∆′1,n,∆
′
2,n),
(iii) (N ′′1 (n), N ′′2 (n)) = (N ′′1 , N
′′
2 )
(
0 −1
1 4
)−3n
+ (∆′′1,n,∆
′′
2,n),
where ∆i,n,∆
′
i,n,∆
′′
i,n ∈ End
(
H3(X∗bo ,Q)
)
and satisfy ∆i,n|W2 = ∆′i,n|W2 = ∆′′i,n|W2 = 0.
(4) The following relations glue the nilpotent cones in (2) (see Fig. 3):
N1(n) = N ′′1 (n), N2(n) = N ′2(n), N ′′2 (n) = N ′1(n− 1).
Proof. (1) Recall that ρ̌ is defined by ρ̌ = ϕ̌13ϕ̌32ϕ̌21. Each isomorphism ϕ̌ij preserves the
monodromy weight filtrations defined for each boundary point ok (see Proposition 4.4). Hence,
ρ̌ : H3(Xbo ,Q)→ H3(Xbo ,Q) preserves the monodromy weight filtration as claimed.
(2) We have introduced the generators of the nilpotent cones Σ
(n)
ok by Ni(n), N ′i(n) and N ′′i (n)
for k = 1, 2, 3, respectively, in Section 4.2.3. Then the claim follows from the properties (3)
and (4) (see also Fig. 3).
(3) By the definition of Ni(n), it is straightforward to calculate Ni(1) as
Ni(1) = ρ̌−1Niρ̌ =
{
−4N1 + 15N2 + ∆1,1, i = 1,
−15N1 + 56N2 + ∆2,1, i = 2,
where ∆i,1 satisfy ∆1,1|W2 = ∆2,1|W2 = 0 on the subspace W2 ⊂ H3(X∗bo ,Q). We note the
relation
(
0 −1
1 4
)3
=
(−4 −15
15 56
)
and arrange the above relation into the claimed matrix form for
n = 1. Then we can obtain the claimed formula (i) for general n (in the first line) by evaluating
(ρ̌−nN1ρ̌
n, ρ̌−nN2ρ̌
n) inductively. In the evaluation, we should note that ρ̌−1∆i,n−1ρ̌|W2 = 0 if
∆i,n−1|W2 = 0 since ρ̌ preserves the monodromy weight filtration. For the second formula (ii),
we note the relation
(N ′1, N
′
2) = (N1, N2)
(
−1 0
4 1
)
+ (∆′1,0, 0)
obtained in Proposition 4.9(1). Taking the conjugations ρ̌−n(-)ρ̌n on the both sides of this
relation, and using the first formula (i) for ρ̌−n(N1, N2)ρ̌n, we have the claimed formula. In the
derivation, we use the relation(
0 −1
1 4
)3n(−1 0
4 1
)
=
(
−1 0
4 1
)(
0 −1
1 4
)−3n
and also the property ρ̌−1∆′i,n−1ρ̌|W2 = 0 if ∆′i,n−1|W2 = 0. For the third relation (iii), calcula-
tions are similar but we need to use the relation
(
0 −1
1 4
)3n ( 1 4
0 −1
)
=
(
1 4
0 −1
) (
0 −1
1 4
)−3n
.
(4) Since Ni(n), N ′i(n), N ′′i (n) are defined by the conjugation of Ni(n − 1), N ′i(n − 1) and
N ′′i (n− 1) by ρ̌, it is sufficient to show the equalities
N1 = N ′′1 , N2 = N ′2, N ′′2 (1) = N ′1.
The first two relations are verified already in Proposition 4.9 and (4.11). For the last relation,
we evaluate N ′′2 (1) = ρ̌−1N ′′2 ρ̌ directly verifying its equality to N ′1. �
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 19
Figure 3. Nilpotent cones glued in End
(
H3(X∗bo ,R)
)
. Glueing continues to the both infinities, corre-
sponding to the monodromy actions ρ̌n, n→ ±∞. This should be compared with Fig. 1.
Corollary 4.12. Consider the left ideal I2 :=
{
X ∈ End
(
H3(X∗b0 ,R)
)
|X|W2 = 0
}
of
End
(
H3(X∗b0 ,R)
)
, and π : End
(
H3(X∗b0 ,R)
)
→ End
(
H3(X∗b0 ,R)
)
/I2 be the natural projection
as a vector space. Then, taking the closure in End
(
H3(X∗b0 ,R)
)
/I2, we have⋃
n
π
(
ρ̌−n
(
Σo2 ∪ Σo1 ∪ Σo3
)
ρ̌n
)
= R>0c̄1 + R>0c̄2,
where c̄1 = −N̄1 + (2 +
√
3)N̄2 and c̄2 = N̄1 − (2−
√
3)N̄2 with N̄i = π(Ni), i = 1, 2.
Proof. From Proposition 4.11, we have
π (Σo1 ∪ Σo2 ∪ Σo3) = R≥0π(N ′1) + R≥0π(N ′′2 ) = R≥0π(4N2 −N1) + R≥0π(4N1 −N2).
Evaluating the matrix power
(
0 −1
1 4
)3n
, it is easy to see that
lim
n→∞
R≥0π(N1(n)) = lim
n→∞
R≥0π(N2(n)) = R≥0c̄1
and
lim
n→−∞
R≥0π(N1(n)) = lim
n→−∞
R≥0π(N2(n)) = R≥0c̄2.
Then the claim follows from the gluing property (1) of Proposition 4.11. �
4.3 Flopping curves and TE1
The matrix TE1 arises from the tangential intersection of the relevant components of the dis-
criminant Dis in the moduli space Mcpx
X∗ . As noted in the remark above, TE1 may be identified
with the first order correction from the quantum cohomology of X1. To see this, let us introduce
Nf
1 := log
(
T−1
x T 4
y
)
= 4N2 −N1 (4.12)
and Nf
2 = N ′2 = N2. Here, we should note the difference in Nf
1 from the definition N ′1 =
log
(
T−1
E1
T−1
x T 4
y
)
.
Proposition 4.13. Define C ′ijk and Cf
ijk by N ′iN
′
jN
′
k = C ′ijkN0 and Nf
i N
f
jN
f
k = Cf
ijkN0 with N0
as given in Proposition 4.1. Non-vanishing (totally symmetric) C ′ijk and Cf
ijk are given by
(C ′111, C
′
112, C
′
122, C
′
222) = (5, 10, 10, 5),
(Cf
111, C
f
112, C
f
122, C
f
222) = (−45, 10, 10, 5). (4.13)
20 S. Hosono and H. Takagi
Proof. We derive these numbers by direct calculations of matrix products. �
The nilpotent matrices N ′1, N ′2 follow from the B-structure at o2, which has been identified
with the A-structure of X2. Hence the first equality in (4.13) is a consequence from mirror
symmetry. To see more details of the equality, let us recall the so-called mirror map which are
defined by
t′i =
∫
A′i
Ωx′∫
A′0
Ωx′
, ti =
∫
Ai
Ωx∫
A0
Ωx
(4.14)
for each boundary point o2 and o1, respectively. If we relate these local definitions by the
isomorphism ( tϕb′obo)−1 : H3(X∗bo ,Z) → H3(X∗b′o ,Z) along the path pb′o←bo (cf. Proposition 4.4),
we have
t′1 = −t1, t′2 = 4t1 + t2.
Proposition 4.14. Let Cijk be as defined in Proposition 4.1. Also set q′1 := et
′
1 and q1 = et1.
Then we have the following relations
Cf
ijk =
∑
l,m,n
Clmn
dtl
dt′i
dtm
dt′j
dtn
dt′k
and
C ′111 + 50
q′1
1− q′1
= Cf
111 + 50
q1
1− q1
(
dt1
dt′1
)3
. (4.15)
Proof. It is easy to verify these. For the second relation, we note that 50 q1
1−q1
(
dt1
dt′1
)3
= 50 +
50
q′1
1−q′1
for q1 = 1/q′1. �
The equality (4.15) is a consequence of the flop invariance of the quantum cohomology (see,
e.g., [29, 30, 37]). As mentioned in Remark 4.7, the number 50 represents the flopping curves.
Comparing this with Proposition 4.13, we see that the monodromy TE1 encodes the data of the
flopping curves which is in the first order of the quantum cohomology of X1.
4.4 Prepotentials
The flop invariance expressed in (4.15) is known more precisely as the invariance of quantum
cohomology under analytic continuations, where all higher order quantum corrections are taken
into account. Here we rephrase this property as a property of the so-called prepotentials.
For the B-structure at each boundary point, we can define the prepotential. For example for
the B-structure at o1 and o2, respectively, they are given by
F =
1
2
3∑
i=0
∫
Ai
Ωx
∫
Bi
Ωx, F ′ = 1
2
3∑
i=0
∫
A′i
Ωx′
∫
B′i
Ωx′
with the symplectic integral bases for period integrals in Π(x, y) and Π′(x′, y′).
Proposition 4.15. By the isomorphism ( tϕb′obo)−1 : H3(X∗bo ,Z) → H3(X∗b′o ,Z) along the path
pb′o←bo chosen as in Proposition 4.4, F and F ′ are related by
F ′ = F +
1
2
2∑
i,j=1
Qij
∫
Ai
Ωx
∫
Aj
Ωx,
where (Qij) =
(−25 −2
2 0
)
.
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 21
Proof. Using the basis {Ai, Bj}, {A′i, B′j}, the connection matrix has the form
(
tϕb′ob0
)−1
=
−1 0 0 0 0 0
0 1 0 0 0 0
0 −4 −1 0 0 0
0 2 0 −1 0 0
0 −25 −2 −4 1 0
0 0 0 0 0 −1
,
which gives the analytic continuation by Π′(x′, y′) = ( tϕb′ob0)−1Π(x, y). From this, we read
A′0 = −A0, B′0 = −B0 and(
A′1
A′2
)
= R
(
A1
A2
)
,
(
B′1
B′2
)
=
(
tR
)−1
(
B1
B2
)
+
(−25 −2
2 0
) (
A1
A2
)
,
where we set R =
(
1 0
−4 −1
)
. The claimed formula is immediate from these. �
As we can deduce in the above proof, the prepotentials are invariant only up to quadratic
terms of the Ai-periods under the analytic continuations even if they are symplectic and also
preserve the monodromy weight filtrations. However the so-called Yukawa couplings are invariant
since they are given by the third derivatives of the prepotentials with respect to the coordinates ti
(see (4.14)).
5 Gluing monodromy nilpotent cones II
We will study the following Calabi–Yau threefold of complete intersections:
X =
(
P3| 2 1 1
P3| 2 1 1
)2,66
.
We assume the defining equations of X are chosen general unless otherwise mentioned. For
such X, there is no other birational model than X. However X has an interesting birational
automorphism of infinite order [41], and also has a non-trivial movable cone similar to the one
in the preceding section.
5.1 Birational automorphisms of infinite order
Let πi : X → P3 be the projections to the first and second factor of P3 × P3 for i = 1 and 2,
respectively. It is easy to see that the projection πi is surjective and generically 2 : 1. We
consider the Stein factorization X → Wi → P3 of the morphism πi : X → P3 and denote the
morphism by φi : X →Wi for i = 1, 2.
Proposition 5.1. For i = 1, 2, the morphism Wi → P3 is a double cover of P3 branched along
an octic, and Wi is a (smooth) Calabi–Yau threefold.
Proof. We omit proofs since they are standard (see, e.g., [41]). �
Let τ̃i : Wi ' W+
i be the deck transformation of the covering Wi → P3. Then we have the
map τi which covers τ̃i as in the following diagram:
XX X
P3 P3
W2W+
2 W1 W+
1 .
τ̃1τ̃2 ''
∼τ1∼τ2
φ1φ2
oo //
π2
��
π1
��
��
��
~~
2:1 %%
��
��
2:1yy
Proposition 5.2. The following hold:
22 S. Hosono and H. Takagi
(i) The map τi : X 99K X is birational but not bi-holomorphic.
(ii) The morphism φi : X → Wi contracts 80 lines and 4 conics to points, and the birational
map τi is an Atyah’s flop of these curves.
Proof. (i), (ii) See the reference [41, Proposition 6.1]. �
Proposition 5.3. (1) Bir(X) = Aut(X) · 〈τ1, τ2〉. (2) τ2
i = id for i = 1, 2. Also τ1τ2 has infinite
order.
Proof. See [41, Lemma 6.4]. �
5.2 Mirror family of X
We can describe the mirror family X∗ →Mcpx
X∗ of X by writing X in terms of a Gorenstein cone
following Batyrev–Borisov. The parameter space of the defining equations up to isomorphisms
naturally gives the moduli space Mcpx
X∗ , which turns out to be compactified to P2 as before.
Here we will not go into the details of the mirror family, but we only write the form of the
Picard–Fuchs differential operator in the affine coordinate [1, x, y] ∈Mcpx
X∗ = P2.
Proposition 5.4. Picard–Fuchs equations of the family on the affine coordinate [1, x, y] are
given by D1w(x, y) = D2w(x, y) = 0 with
D1 =
(
3θ2
x − 4θxθy + 3θ2
y
)
− (θx + θy)(2θx + 2θy − 1)(10x+ 6y)
+ 4θx(2θx + 2θy − 1)(x− y),
D2 =
(
θ3
x − θ2
xθy + θxθ
2
y − θ3
y
)
− 2(θx + θy)
2(2θx + 2θy − 1)(x− y),
where θx = x ∂∂x , θy = y ∂∂y . The discriminant locus of this system is given by Dis = D1 ∪D2 ∪
D3 ∪Dis0 with
Dis0 =
{
(1− 4x− 4y)4 − 128xy
(
17 + 56(x+ y) + 16
(
x2 + y2
))
= 0
}
,
and the coordinate lines Di of P2.
Proof. The differential operators D1 and D2 arise from the Gel’fand–Kapranov–Zelevinski sys-
tem after finding suitable factorizations of differential operators. See [22] for more details.
Once D1 and D2 are determined, it is straightforward to determine the discriminant locus (sin-
gular locus) of the system from the equations of the characteristic variety. �
From the forms of D1and D2, the origin x = y = 0 is expected to be a LCSL. In fact, we can
verify all the properties for the LCSL in Definition 2.1. We also verify that there is no other
LCSL point in Mcpx
X∗ = P2. In Fig. 4, we schematically describe the structure of the moduli
space Mcpx
X∗ . There, as in the preceding example, we see that the component Dis0 intersects
tangentially with the divisors D1 = {x = 0} and D2 = {y = 0}. This time, we blow-up at these
two intersection points successively four times to make the intersections normal crossing (see
Fig. 4 right).
Remark 5.5. As in the previous example, we should be able to arrive at the mirror family
X∗ →Mcpx
X∗ starting with a special family {Xsp}a,b. But we leave this task for other occasions,
since we have the mirror family in any case as above.
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 23
Figure 4. Blowing-up Mcpx
X∗ = P2. There is only one LCSL at o1 in this case, and we blow-up four
times at the two points introducing exceptional divisors E1 and E2 shown.
5.3 B-structure of X at the origin o
As in Section 4.1, the canonical integral and symplectic structure can be introduced from the
power series solution
w0(x, y) =
∑
n,m
Γ(1 + 2n+ 2m)Γ(1 + n+m)2
Γ(1 + n)4Γ(1 +m)4
xnym
around the origin o := [1, 0, 0]. Simply replacing the necessary parameters in the general formula
[26, Section 6.3.1], and fixing the so-called quadratic ambiguities there by Ckl = 0, we obtain
the canonical integral and symplectic structure in the form of period integrals Π(x, y) with the
corresponding symplectic basis
{A0, A1, A2, B2, B1, B0} ⊂ H3(X∗bo ,Z) with Ai ∩Bj = δij , Ai ∩Aj = Bi ∩Bj = 0,
where a base point bo is taken near the origin. We denote by Tx the matrix of the monodromy
transformation of Π(x, y) for a small loop around the divisor D1 = {x = 0}, and similarly
denote by Ty for a small loop around D2 = {y = 0}. Writing the local solutions explicitly, it is
straightforward to have
Tx =
1 0 0 0 0 0
1 1 0 0 0 0
0 0 1 0 0 0
3 6 6 1 0 0
1 2 6 0 1 0
−4 −1 −3 0 −1 1
, Ty =
1 0 0 0 0 0
0 1 0 0 0 0
1 0 1 0 0 0
1 6 2 1 0 0
3 6 6 0 1 0
−4 −3 −1 −1 0 1
.
We introduce the dual basis B = {αi, βi} of H3(X∗bo ,Z) and consider the dual actions Tx :=
( tTx)−1 and Ty := ( tTy)
−1 on H3(X∗bo ,Z) which are clearly unipotent.
Definition 5.6. We define the monodromy nilpotent cone at o by
Σo =
{∑
λiNi |λi ≥ 0
}
⊂ End
(
H3(X∗bo ,Z)
)
with N1 = − t(log Tx) and N2 = − t(log Ty).
Using the explicit forms of these matrices, we verify the following properties:
Proposition 5.7.
(1) The nilpotent element Nλ :=
∑
i λiNi, λi > 0, defines the weight monodromy filtration
W0 ⊂W2 ⊂W4 ⊂W6 = H3(X∗b0 ,Z) with the same form W2i as given in (4.4).
24 S. Hosono and H. Takagi
(2) We have
NiNjNk = CijkN0
with totally symmetric Cijk defined by C111 = C222 = 2, C122 = C112 = 6, Cijk = 0
otherwise, and N0 =
(
0 1
O5 0
)
where O5 is the zero matrix of size 5× 5.
From the above proposition, we see that the origin satisfies the conditions for LCSL. Also,
looking other boundary points in the moduli space Mcpx
X∗ , we see that no other LCSL exists
in Mcpx
X∗ .
5.4 Gluing the monodromy nilpotent cone Σo
Although there is only one LCSL in the mirror family X∗ →Mcpx
X∗ , we can find the monodromy
transformations which correspond to the birational automorphisms τ1 and τ2 of X. We observe
that the monodromy nilpotent cone Σo extends to a larger cone (or cone structure) using these
monodromy transformations, and we will identify the resulting cone structure with the movable
cone Mov(X) of X.
5.4.1 Path po←Ei←o, i = 1,2
As shown in Fig. 4, the discriminant locus Dis has non-normal crossing intersection at three
points. To make the intersections normal, we blow-up successively four times at the two points
near the origin o. We denote by E1 and E2, respectively, the exceptional divisors introduced
by the blow-ups (see Fig. 4 right). As we see in the form of the discriminant Dis, the family
over Mcpx
X∗ is symmetric under x ↔ y. Because of this symmetry reason, it suffices to describe
the divisor D2 = {y = 0} which intersects with E1 at [1, x, y] =
[
1, 1
4 , 0
]
∈ P2. Explicitly, we
introduce the blow-up coordinate at the origin q12 := E1 ∩Dy by
s1 = 4x− 1, s2 =
1
26
y
(1− 4x)4
.
Definition 5.8. Let R12 =
{
1
4 + s1
4 e
iθ | 0 ≤ θ ≤ 2π
}
be a small loop around E1 on D2. We
denote by pq12←bo = {(1− t)bo + tq12 | 0 ≤ t ≤ 1− ε} the straight line connecting the base point
bo near o and a point q12 on the small loop R12. Then we define
pbo←E1←b0 := (pq12←bo)−1 ◦R12 ◦ pq12←bo
to be the composite path which encircles the divisor E1 from the base point bo. In a similar
way, we define a closed path pb0←E2←bo which encircle the divisor E2 from b0 (see Fig. 4).
5.4.2 Monodromy around Ei
Let (x′, y′) =
(
1
x ,
y
x
)
be the affine coordinate with the origin [0, 1, 0] ∈ P2 and b′o be a base
point near the origin. We denote by T′x′ and T′y′ the local monodromy around x′ = 0 and
y′ = 0, respectively. Conjugating T′x′ , T′y′ by the connection matrix for the path pb′o←b0 =
pb′0←q12
◦ pq12←bo , we define the corresponding monodromy matrices Tx′ and Ty′ for loops with
the base point bo. We define Tx′ := (tTx′)
−1 and Ty′ := ( tTy′)
−1 to be the linear actions on the
dual space H3(X∗bo ,Z).
Proposition 5.9. We have
Tx′ =
−1 −1 3 6 −10 2
0 1 −6 −12 8 2
0 0 −1 0 12 −6
0 0 0 −1 −6 3
0 0 0 0 1 −1
0 0 0 0 0 −1
, Ty′ =
1 0 −1 1 3 4
0 1 0 −6 −6 −3
0 0 1 −2 −6 −1
0 0 0 1 0 1
0 0 0 0 1 0
0 0 0 0 0 1
.
In particular we have Ty′ = Ty.
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 25
Proof. These are based on explicit calculations. Here we only sketch the calculations. We
first make local solutions using the coordinate (s1, s2) centered at q12. Then their domain of
convergence have overlap both with the local solutions around (x, y) = (0, 0) and (x′, y′) = (0, 0).
Then it is straightforward to obtain the connection matrices. The local monodromy matrices T′x′
and T′y′ are easily read off from the local solutions. Then by conjugating these local monodromy
matrices by the connection matrix, we have the expressions for Tx′ , Ty′ as the linear actions on
H3(X∗b0 ,Z). Translating these to H3(X∗bo ,Z), we obtain Tx′ and Ty′ . �
Similarly we define the monodromy matrix TE1 along the loop pbo←E1←bo and set TE1 :=
( tTE1)−1. Corresponding to Proposition 4.6 we have
Proposition 5.10 (‘Picard–Lefschetz formula’ for the flopping curves).
(1) The monodromy matrix is given by
TE1 =
−1 0 0 0 0 0
0 1 −6 0 48 0
0 0 −1 0 0 0
0 0 0 −1 −6 0
0 0 0 0 1 0
0 0 0 0 0 −1
.
In particular, this is quasi-unipotent.
(2) For T 2
E1
we have
T 2
E1
=
1
1 96
1
1
1
1
, i.e.,
α1 → α1 + 96β1,
β1 → β1,
αi = αi, βi = βi, i 6= 1.
(3) By symmetry, we have similar formula for TE2 and T 2
E2
. In particular, T 2
E2
is given by
α2 → α2 + 96β2, β2 → β2, with αi = αi, βi = βi for i 6= 2.
Proof. These results follow from making local solutions and the analytic continuations of them.
Again, calculations are straightforward since local solutions around (x, y) = (0, 0) and (s1, s2) =
(0, 0) have overlap in their domains of convergence. �
Remark 5.11. (1) As before, the monodromy action (2) in the above proposition is expressed
in terms of the symplectic basis {Ai, Bj} of H3(X∗bo ,Z) as
A1 → A1, B1 → B1 − 96A1.
(2) We have seen in Proposition 5.2 that each τi : X 99K X is an Atiyah’s flop with respect
to 80 lines and also 4 conics. We observe that 96 = 80 + 4× 22 holds for the number in T 2
Ei
. We
can verify the corresponding relations also for other examples. Based on these, we conjecture
the following general form:
A1 → A1, B1 → B1 −
(
n0(1) + n0(2)× 22
)
A1
for the Atiyah’s flops of n0(1) lines and n0(2) conics associated to the contractions to the double
cover of P3.
26 S. Hosono and H. Takagi
5.4.3 Monodromy relations
Take affine coordinates (x, y), (x′, y′) and (x′′, y′′) of P2 as shown in Fig. 4. Let Tx′ , Ty′ be as
defined in Proposition 5.9.
Proposition 5.12. The following monodromy relations holds
Tx′ = T−1
E1
T−1
x T 3
y , Ty′ = Ty, TE1Ty = TyTE1 . (5.1)
Proof. We have the second and the third relations since all the divisors are normal crossing
after the blow-ups. We can verify the first relation directly by using Tx = ( tTx)−1, Ty = ( tTy)
−1
given in Section 5.3 and Tx′ , TE1 in Section 5.4.2. �
Definition 5.13. Define the following conjugations of Tx, Ty by TE1 :
T̃x := T−1
E1
TxTE1 , T̃y := T−1
E1
TyTE1 .
Using these, we define the monodromy nilpotent cone by
Σ̃o :=
{∑
λiÑi |λi > 0
}
⊂ End
(
H3(X∗bo ,R)
)
,
where Ñ1 := log T̃x and Ñ2 := log T̃y.
Proposition 5.14. The (closures of the) monodromy nilpotent cones Σo and Σ̃o glue along the
ray R≥0N2, but they are not on the same two dimensional plane.
Proof. Using the monodromy relations in Proposition 5.12, we have T̃y′ = Ty. Hence the claim
is immediate since we have Ñ2 = N2 by definition. To see the second claim, we use again the
monodromy relations to have
T̃x = T−1
E1
TxTE1 = T−1
E1
T 3
y T
−1
x′ = T−1
E1
T−1
x′ T
3
y ,
which is reminiscent of the relation (4.9). In fact, after some matrix calculations, we obtain
Ñ1 = 6N2 −N1 + ∆1, ∆1 =
0 0 0 0 48 − 44
3
0 0 0 0 −112 48
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
, (5.2)
where ∆1 satisfies ∆1|W2 = 0. Since the nilpotent cone Σo lies on the plane spanned by N1
and N2, and aN1 + bN2|W2 6= 0 holds for any a, b, the basis element Ñ1 does not lie on the same
plane as Σo. �
5.4.4 Gluing nilpotent cones
As the example in the previous section, the structure of the moduli space Mcpx
X∗ is symmetric
under the exchange of x and y. Hence, corresponding to (5.1), we have
Tx′′ = Tx, Ty′′ = T−1
E2
T−1
y T 3
x , TE2Tx = TxTE2 . (5.3)
When we define T̃ ′x := T−1
E2
TxTE2 , T̃ ′y := T−1
E2
TyTE2 , we have the following relations
Ñ ′1 = N1, Ñ ′2 = 6N1 −N2 + ∆′1
for Ñ ′1 := log T̃x, Ñ ′2 := log T̃y with ∆′1|W2 = 0. This entails the corresponding gluing property
described in Proposition 4.9. We summarize these two actions into the following general form.
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 27
Definition 5.15. We denote by τEi the conjugations by TEi on End
(
H3(X∗bo ,Q)
)
, which act
on the nilpotent matrices N in general as
τE1(N) = T−1
E1
NTE1 , τE2(N) = T−1
E2
NTE2 .
We set G := 〈τE1 , τE2〉, i.e., the group generated by τE1 and τE2 .
Proposition 5.16.
(1) The actions of τnEi
∈ G on N1 = log Tx, N2 = log Ty are summarized as
(
τnE1
(N1), τnE1
(N2)
)
= (N1, N2)
(
−1 0
6 1
)n
+ (∆n, 0),
(
τnE2
(N1), τnE2
(N2)
)
= (N1, N2)
(
1 6
0 −1
)n
+ (0,∆′n),
where ∆n, ∆′n are elements in End
(
H3(X∗bo ,Q)
)
satisfying ∆n|W2 = ∆′n|W2 = 0. In
particular, we have
τnE1
(N2) = N2, τnE2
(N1) = N1.
(2) The action of σ ∈ G on ∆n,∆
′
n preserves the vanishing properties of ∆n, ∆′n on W2, i.e.,
σ(∆n)|W2 = σ(∆′n)|W2 = 0.
(3) ∆n, ∆′n have the following forms:
∆2m =
(
O24
−96m 0
0 −96m
O44 O42
)
, ∆2m−1 =
O24
96
(
m− 1
2
)
− 44
3
−122 96
(
m− 1
2
)
O44 O42
and ∆′n = p23p45∆np23p45, where Oab is the a × b zero matrix and pij represents the
permutation matrix for the transposition (i, j).
Proof. These properties are verified by explicit calculations using the matrix representations
Tx, Ty and TEi given previous sections. The vanishing properties follow inductively from ∆1|W2 =
∆′1|W2 = 0 and the fact that both TE1 and TE2 preserve the monodromy weight filtration
W0 ⊂W2 ⊂W4 ⊂W6 = H3(X∗bo ,Q). �
As before, let I2 :=
{
X ∈ End
(
H3(X∗b0 ,R)
)
|X|W2 = 0
}
be an left ideal of End
(
H3(X∗b0 ,R)
)
,
and π : End
(
H3(X∗b0 ,R)
)
→ End
(
H3(X∗b0 ,R)
)
/I2 be the natural projection. Since TEi preserve
the monodromy weight filtration, and by the definition of τEi , it is easy to see that σ(I2) ⊂ I2 for
all σ ∈ G. Hence we have the naturally induced G action on the quotient End
(
H3(X∗b0 ,R)
)
/I2
by σ(X + I2) := σ(X) + I2. Note that, if we denote by σ̄ the action of σ ∈ G on the quotient
space, we have στ = τ̄ σ̄ (i.e., anti-homomorphism by our convention for the adjoint action) for
all σ, τ ∈ G.
Corollary 5.17. Denote by N̄i := π(Ni) the basis of the cone π(Σo) in the quotient space. Then
the following hold:
(1) Define τ12 := τ1τ2 with τi = τEi. We have
(
τ̄n12(N̄1), τ̄n12(N̄2)
)
=
(
N̄1, N̄2
)(35 6
−6 −1
)n
.
(2) {σ ∈ G | σ̄(N̄i) = N̄i, i = 1, 2} = 〈τ2
1 , τ
2
2 〉.
28 S. Hosono and H. Takagi
Figure 5. Gluing nilpotent cones. The nilpotent cones σ(Σo) glue in End
(
H3(X∗bo ,R)
)
. The com-
posite actions of τ21 and τ22 on each ray are not trivial, although they are trivial on the images in
End
(
H3(X∗bo ,R)
)
/I2.
(3) Taking the closure in End
(
H3(X∗b0 ,R)
)
/I2, we have⋃
σ∈G
π(σ(Σo)) = R>0
(
−N̄1 + (3 + 2
√
2)N̄2
)
+ R>0
(
N̄1 + (3− 2
√
2)N̄2
)
.
Proof. The equality (1) follow from Proposition 5.16(1) and τ̄12 = τ̄2τ̄1. By definition, G
is generated by τ1, τ2. Then the claim (2) follows from Proposition 5.16(1) and the above
equality (1). To show the claim (3), we write by (N1, N2)>0 the cone generated by N1 and N2.
Then we first show that the following cones successively glue together to a large cone:(
τn12(N1), τn12(N2)
)
>0
,
(
τn12(τ1N1), τn12(τ1N2)
)
>0
, n ∈ Z.
Using the property τ1(N2) = N2, τ2(N1) = N1, we have
τn12(τ1N1) = τn+1
12 (N1), τn12(τ1N2) = τn12(N2),
by which we can arrange a sequence of cones schematically as follows:(
τn+1
12 (τ1N1), τn+1
12 (τ1N2)
)
>0
(
τn12(τ1N1), τn12(τ1N2)
)
>0
· · ·
q q q q q
· · ·
(
τn+1
12 (N2), τn+1
12 (N1)
)
>0
(
τn12(N2), τn12(N1)
)
>0
Let us note that τ2τ1 = τ2
2 τ
−1
12 τ
2
1 and τ2 = τ2
2 τ
−1
12 τ1 hold. Then, using these relations, we can
deduce the decomposition
G =
〈
τ12, τ
2
1 , τ
2
2
〉
∪
〈
τ12, τ
2
1 , τ
2
2
〉
τ1.
Since τ2
1 , τ2
2 have trivial actions on N̄i, i = 1, 2, the above sequence of the cones explain the union⋃
σ∈G π(σ(Σo)). After some linear algebra of the matrix power
(
35 6
−6 −1
)n
, we can determine the
infinite union in the claimed form. �
5.5 Flopping curves and TE1
The monodromy TE1 has appeared in the moduli space from the tangential intersection of the
discriminants. This is quite parallel to Section 4.3. However, TE1 is not unipotent but only
quasi-unipotent in the present case. This prevents a parallel definition to the second equation
in (4.12), but this time we set
Nf
1 := 6N2 −N1
with Ñ1 = Nf
1 + ∆1 (see (5.2)) and also Ñ2 = Nf
2 = N2. Then we have
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 29
Proposition 5.18. Let ÑiÑjÑk = C̃ijkN0 and Nf
i N
f
jN
f
k = Cf
ijkN0 with N0 as given in Proposi-
tion 4.1. Non-vanishing (totally symmetric) C̃ijk and Cf
ijk are given by(
C̃111, C̃112, C̃122, C̃222
)
= (2, 6, 6, 2),(
Cf
111, C
f
112, C
f
122, C
f
222
)
= (−110, 6, 6, 2). (5.4)
As before, the first equality of (5.4) is explained by mirror symmetry, i.e., the isomorphism
of the B-structure at o with the A-structure of X. To see this isomorphism more explicitly, we
recall the mirror map
ti =
∫
Ai
Ωx∫
A0
Ωx
defined by the B-structure at the LCSL o. The monodromy matrix TE1 = ( tTE1)−1 represents
the isomorphism H3(X∗bo ,Z) → H3(X∗b′o ,Z) which follows from the analytic continuation of the
period integral Π(x, y) along the path pb0←E1←bo . After the continuation, the coordinate (t1, t2)
transformed to (t′1, t
′
2) with
t′1 = −t1, t′2 = 6t1 + t2.
Corresponding to Proposition 4.14, we now have
Proposition 5.19. Let Cijk be as defined in Proposition 4.1. Also set q′1 := et
′
1 and q1 = et1.
Then we have the following relations
Cf
ijk =
∑
l,m,n
Clmn
dtl
dt′i
dtm
dt′j
dtn
dt′k
and
C̃111 + 80
q′1
1− q′1
+ 4
23q′21
1− q′21
= Cf
111 +
(
80
q1
1− q1
+ 4
23q2
1
1− q2
1
)(
dt1
dt′1
)3
. (5.5)
In the above equality, we see the invariance of the quantum cohomology of X under birational
transformations. We note that the equality (5.5) has a slightly more general form than the
familiar form (4.15) due to the existence of 4 conics in the flopping curves.
6 Summary and discussions
We have studied gluings of monodromy nilpotent cones through monodromy relations coming
from boundary divisors. Under the mirror symmetry, we have identified them with the corre-
sponding gluings along codimension-one walls of the Kähler cones in birational geometry. In this
paper, we confined ourself to two specific examples by doing explicit monodromy calculations.
However, it is naturally expected that the observed gluings of monodromy nilpotent cones and
their interpretation in mirror symmetry hold in general.
We present below some discussions and related subjects in order. In particular, we briefly
report the gluing in the case of K3 surfaces whose moduli spaces have parallel structures to the
Calabi–Yau threefolds X and X∗ studied in Sections 3 and 4.
6.1. The gluing of monodromy nilpotent cones has been done naturally through the mon-
odromy relations (4.9), (4.10) and also (5.1), (5.3). These relations came from boundary divisors
which have tangential intersections with some component of discriminant and the blowing-ups
at the intersection points. As remarked in Remarks 4.7, 5.11, these tangential singularities are
30 S. Hosono and H. Takagi
related to the contractions in the birational geometry of the mirror Calabi–Yau manifolds. We
expect some generality in the degenerations of the mirror families X∗ when we approach to the
exceptional divisors Ei of the blow-ups. We have to leave this for future investigations although
we note that a categorical study of the mirror symmetry for conifold transitions has been put
forward in a recent work [11].
6.2. In the homological mirror symmetry due to Kontsevich [34], monodromy transformations
in B-structures are interpreted as the corresponding transformations in the derived category of
coherent sheaves Db(X). From this viewpoint, the gluing of nilpotent cones in End
(
H3(X∗,Z)
)
suggests the corresponding gluing of Kähler cones in End(K(X)) as a homological extension of
the movable cones. The resulting wall structures of the gluing in End(K(X)) should be regarded
as the wall structures in the stability space [8] of the objects in Db(X).
6.3. As addressed in Remark 3.4, one can expect non-trivial birational geometry also for other
examples of complete intersections described by Gorenstein cones [3]. Among such examples,
there are complete intersections whose projective geometry fits well to the so-called linear duality
(see Appendix B). We have for example the following complete intersection:
X =
(
P4| 2 1 1 1
P3| 1 1 1 1
)2,56
,
which shares many properties with (3.1) in Section 3, e.g., three birational models come together
when we construct the complete intersection of the form X. Although we do not have birational
automorphisms of infinite order in this example, these three birational models are explained
nicely by “double linear duality”, a certain composite of two different linear dualities. We will
report this elsewhere.
6.4 (Cayley model of Reye congruences). Historically the Calabi–Yau complete inter-
section studied in Section 3 is a generalization of the following K3 surface:
X =
(
P3 | 1 1 1 1
P3 | 1 1 1 1
)
,
which is called a Cayley model of Reye congruences. When we take the defining equations
general, X is a smooth K3 surface of the Picard lattice isomorphic to M :=
(
Z2, ( 4 6
6 4 )
)
. This K3
surface has been studied in [13, 42] as an example which has an automorphism ρ of infinite
order and also positive entropy. Actually, we have the same diagram as (3.4) with the parallel
definitions of Xi (X1 := X) and Zi as well as ρ in Proposition 3.8. The difference is in that
all Xi and Zi are smooth K3 surfaces and hence isomorphic to each other under the morphisms,
e.g., πij and ϕij . For K3 surfaces, we have the so-called counting formula [24] for the number
of Fourier–Mukai partners. Based on it, it is easy to see that the set FM(X) of Fourier–Mukai
partners consists of only X itself.
The construction of the mirror family of X is similar to Section 3.3, and there appear three
LCSL oi, i = 1, 2, 3, on the compactified moduli space Mcpx
X∗ = P2. As before, we determine
the connecting matrices ϕ̌ij by blowing-up at three points with (fourth) tangential intersections
(cf. Fig. 2). Making similar canonical bases of period integrals as in (4.1) at each point, which
represents bases of the transcendental lattice TX∗ ' U ⊕M of the mirror K3 surface X∗, we
obtain
ϕ̌21 =
(−1 0 0 0
0 1 −3 0
0 0 −1 0
0 0 0 −1
)
, ϕ̌32 =
(−1 0 0 0
0 −3 1 0
0 −1 0 0
0 0 0 −1
)
, ϕ̌13 =
(−1 0 0 0
0 −1 0 0
0 −3 1 0
0 0 0 −1
)
,
ρ̌ := ϕ̌13ϕ̌32ϕ̌21 =
(−1 0 0 0
0 3 −8 0
0 8 −21 0
0 0 0 −1
)
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 31
as elements in O(U⊕M,Z). Here we define U = Ze⊕Zf to be the hyperbolic lattice
(
Z2, ( 0 1
1 0 )
)
and order the bases of U ⊕M as Ze⊕M ⊕ Zf when writing the above matrix forms.
The classical mirror symmetry summarized in Section 2 applies to the so-called (families of)
lattice polarized K3 surfaces replacing the Kähler cone with the ample cones [10]. In our case
here, we consider a primitive lattice embedding M ⊕ U ⊕ M̌ ⊂ LK3 with a fixed decomposition
M⊥ = U ⊕ M̌ . Then X is a member of the M -polarized K3 surfaces, while the mirror X∗ is
a member of M̌ -polarized K3 surfaces (whose transcendental lattice is M̌⊥ = U ⊕M). The
classical mirror symmetry in this case may be summarized in the following isomorphism:
V +
M +
√
−1M ⊗ R ' Ω+(U ⊕M)
for the period domain Ω+(U ⊕ M) = {[ω] ∈ P((U ⊕M)⊗ C) |ω.ω = 0, ω.ω̄ > 0}+ where we
take one of the connected components, and the corresponding component of the tube domain
V +
M = {v ∈M ⊗ R | (v, v)M > 0}+.
Since there are no elements with (v, v)M = −2 in M , the ample cones of general members of
M -polarized K3 surfaces coincide with the positive cone, which is isomorphic to V +
M . Similarly
to what we described in Section 3.2, by gluing the cone R≥0H1 + R≥0H2 ⊂ H2(X,R) by the
morphisms ϕij , we arrive at the positive cone V +
M which is an irrational cone (see [42] and [13,
Section 1.5]). This gluing exactly matches to the gluing the monodromy nilpotent cones at each
boundary point oi by the connection matrix ϕ̌ij . The monodromy relations play the key roles
for the gluing, and they follow from the parallel calculations to those in Section 4. For example,
we have
Tx′ = T−1
x T 3
y , Ty′ = Ty, Tx′′ = Tx, Ty′′ = T−1
y T 3
x
corresponding to (4.9) and (4.10), respectively, with
Tx =
(
1 −1 0 −2
0 1 0 4
0 0 1 6
0 0 0 1
)
, Ty =
(
1 0 −1 −2
0 1 0 6
0 0 1 4
0 0 0 1
)
,
Tx′ = ϕ̌−1
21 Txϕ̌21, Ty′ = ϕ̌−1
21 Tyϕ̌21 and Tx′′ = ϕ̌−1
31 Txϕ̌31, Ty′′ = ϕ̌−1
31 Tyϕ̌31.
As in Section 4.2, exceptional divisors E1, E′1 and E′′1 have to be introduced to determine the
connection matrices ϕ̌ij , but it turns out that their monodromies are trivial, i.e., TE1 = TE′ =
TE′′1 = id. Clearly, this is consistent to our interpretation of these monodromies in terms of the
flopping curves (Proposition 4.6) for the case of Calabi–Yau threefolds.
As this example shows, irrational ample cones indicate infinite gluings of the nilpotent cones
in the mirror side. It is natural to expect that the corresponding property holds for the mirror
symmetry of Calabi–Yau threefolds in general with ample cones replaced by movable cones and
the morphisms by birational maps as known in the so-called movable cone conjecture [32, 39].
We have shown in this paper that, in three dimensions, the gluings of monodromy nilpotent
cones encode the non-trivial monodromies TEi which correspond to the flopping curves.
A Proof of Lemmas 3.5 and 3.6
A.1 Proof of Lemma 3.5
Let us consider the projective spaces P(Vi) with Vi ' C5, i = 1, 2. Here we will only present
a proof of (1), but it should be clear how to modify the following setting to show (2).
We start with our discussion with the following exact sequence, which we obtain by tensoring
the Euler sequence of P(V2) with V1:
0→ V1 ⊗OP(V2)(−1)→ V1 ⊗ V2 ⊗OP(V2) → V1 ⊗ TP(V2)(−1)→ 0.
32 S. Hosono and H. Takagi
In the following arguments, we denote this sequence by
0→ E → V1 ⊗ V2 ⊗OP(V2) → (E⊥)∗ → 0
with defining E := V1 ⊗ OP(V2)(−1) and E⊥ := V ∗1 ⊗ ΩP(V2)(1). We also have the following
diagram of a linear duality (cf. [36, Section 8]):
X1 ⊂ P(E) P(E⊥) ⊃ X2
P(V1 ⊗ V2) P(V2) P(V ∗1 ⊗ V ∗2 )⊃ Z3.
}} !! }} !!
Note that P(E) is isomorphic to P(V1)× P(V2), and OP(E)(1) ' OP(V1)×P(V2)(1, 1) since it is the
pull-back of OP(V1⊗V2)(1) by construction. Therefore X1 is a codimension 5 complete intersection
in P(E) with respect to OP(E)(1), and we have OP(E)(1)|X1 = H1 +H2.
We see that
P(E⊥) = {(w,M) |Mw = 0} ⊂ P(V2)× P(V ∗1 ⊗ V ∗2 ), (A.1)
where we consider V ∗1 ⊗ V ∗2 ' Hom(V2, V
∗
1 ) and M is a 5× 5 matrix. Therefore the image Z of
the map P(E⊥) → P(V ∗1 ⊗ V ∗2 ) consists of 5 × 5 matrices of rank ≤ 4, thus Z is so-called the
determinantal quintic. Note that we can write the determinantal quintic Z3 ⊂ P4
λ in Proposi-
tion 3.3 by Z3 = Z∩P4 for a 4-dimensional linear subspace P4 ⊂ P(V ∗1 ⊗V ∗2 ) with identifying P4
with P4
λ. Moreover, the pull-back of Z3 to P(E⊥) is X2.
By a general fact on linear duality (B.3) in Appendix B, we have
OP(E)(1)|X1 +OP(E⊥)(1)|X1 = det E∗ = 5H2, (A.2)
where we denote by OP(E⊥)(1)|X1 the strict transform of OP(E⊥)(1)|X2 and abbreviate the nota-
tion for the pull-back for det E∗. In this appendix, unless stated otherwise, we will write proper
transforms of a divisor by the same symbol omitting the pull-backs by birational maps. Using
this convention, we have OP(E)(1)|X1 = H1 +H2 and also OP(E⊥)(1)|X1 = LZ3 . Then we have
(H1 +H2) + LZ3 = 5H2,
which gives LZ3 = 4H2 −H1. Therefore, restoring the pull-backs by birational maps, we have
ϕ∗21LZ3 = 4H2 −H1, ϕ∗21LZ2 = H2
in N1(X), which determine ϕ∗21(KX2) as claimed.
A.2 Proof of Lemma 3.6
Basic idea is very similar to the linear duality in the previous section. We consider the following
diagram:
P(V ∗1 ⊗ ΩP(V2)(1))
P(V ∗1 ⊗ V ∗2 ).
P(ΩP(V1)(1)⊗ V ∗2 )
'' ww
Claim A.1. P(V ∗1 ⊗ΩP(V2)(1))→ P(V ∗1 ⊗V ∗2 ) and P(ΩP(V1)(1)⊗V ∗2 )→ P(V ∗1 ⊗V ∗2 ) are flopping
contractions onto the common image Z. Moreover, it is of Atiyah type outside the locus in Z
of corank ≥ 2.
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 33
Proof. This is standard since P(V ∗1 ⊗ ΩP(V2)(1)) → P(V ∗1 ⊗ V ∗2 ) and P(ΩP(V1)(1) ⊗ V ∗2 ) →
P(V ∗1 ⊗ V ∗2 ) are the Springer type resolutions of the image Z (see (A.1)). �
As we have seen in the proof of Lemma 3.5, X2 is contained in P(V ∗2 ⊗ΩP(V1)(1)). Similarly,
X3 is contained in P(ΩP(V2)(1) ⊗ V ∗1 ). Indeed, for the 4-dimensional linear subspace P4 ⊂
P(V ∗1 ⊗ V ∗2 ) such that Z3 = Z ∩ P4, X2 and X3 are the pull-backs of P4 to P(V ∗1 ⊗ ΩP(V2)(1))
and P(ΩP(V1)(1)⊗ V ∗2 ), respectively.
Now we take the fiber product
P := P
(
V ∗1 ⊗ ΩP(V2)(1)
)
×P(V ∗1 ⊗V ∗2 ) P
(
ΩP(V1)(1)⊗ V ∗2
)
.
Claim A.2. It holds that P = PP(V1)×P(V2)
(
ΩP(V1)(1)� ΩP(V2)(1)
)
.
Proof. Note that
P =
{
(w,M, z) |Mw = 0, tzM = 0
}
⊂ P(V2)× P(V ∗1 ⊗ V ∗2 )× P(V1).
Thus the fiber of P→ P(V1)×P(V2) over (w, z) is nothing but P((V1/Cw)∗⊗ (V2/Cz)∗) and the
assertion follows. �
Note that the tautological divisor OP(1) of P
(
ΩP(V1)(1)�ΩP(V2)(1)
)
defines a map to P(V ∗1 ⊗
V ∗2 ) and it is the pull-back of OP(V ∗1 ⊗V ∗2 )(1). We will denote it by LP(V ∗1 ⊗V ∗2 ). By the canonical
bundle formula of P(ΩP(V1)(1)� ΩP(V2)(1)), we have
KP = −16LP(V ∗1 ⊗V ∗2 ) +KP(V1)×P(V2) + det
{
ΩP(V1)(1)� ΩP(V2)(1)
}∗
,
where we omit the notation of the pull-backs for KP(V1)×P(V2) and det{ΩP(V1)(1) � ΩP(V2)(1)}∗.
Since KP(V1)×P(V2) = −5LP(V1)−5LP(V2), where LP(V1) and LP(V2) are the pull-backs of OP(Vi)(1)’s
of P(Vi) on the left and right factors of P(V1)×P(V2), respectively, and det{ΩP(V1)(1)�ΩP(V2)(1)}∗
= 4LP(V1) + 4LP(V2), we have
KP = −16LP(V ∗1 ⊗V ∗2 ) − LP(V1) − LP(V2). (A.3)
By the canonical bundle formula of P(V ∗1 ⊗ ΩP(V2)(1)), we have
−KP(V ∗1 ⊗ΩP(V2)(1)) = 20LP(V ∗1 ⊗V ∗2 ).
Pushing forwards (A.3) to P(V ∗1 ⊗ ΩP(V2)(1)), we obtain
−KP(V ∗1 ⊗ΩP(V2)(1)) = 16LP(V ∗1 ⊗V ∗2 ) + LP(V1) + LP(V2).
Therefore we have
LP(V1) + LP(V2) = 4LP(V ∗1 ⊗V ∗2 ). (A.4)
Now, restricting the above construction over the linear subspace P4 ⊂ P(V ∗1 ⊗ V ∗2 ), we have
P|P4
X2 X3,
P4
zz $$
$$ zz
where we denote by P|P4
the restriction of P over P4. Restricting (A.4) to X2, we have
ϕ∗32(MZ1) + LZ2 = 4LZ3 . (A.5)
This is the claimed relation.
34 S. Hosono and H. Takagi
Corollary A.3. P(V ∗1 ⊗ ΩP(V2)(1)) 99K P(ΩP(V1)(1) ⊗ V ∗2 ) is the flop. Similarly, X2 99K X3 is
the flop.
Proof. Note that LP(V1) and LP(V2) are relatively ample for P(ΩP(V1)(1) ⊗ V ∗2 ) → P(V1) and
P(V ∗1 ⊗ ΩP(V2)(1)) → P(V2), respectively. Since LP(V ∗1 ⊗V ∗2 ) is the pull-backs of a divisor on
P(V ∗1 ⊗ V ∗2 ), we see that −LP(V1) is relatively ample for P(V ∗1 ⊗ ΩP(V2)(1)) → P(V2) by (A.4).
Therefore P(V ∗1 ⊗ ΩP(V2)(1)) 99K P(ΩP(V1)(1) ⊗ V ∗2 ) is the flop. We can show the assertion for
X2 99K X3 in the same way using (A.5). �
B Linear duality
Having the case W = V1⊗V2 and B = P(V2) in mind, we consider the exact sequence of sheaves
(vector bundles) in the following general form with dimW = N :
0→ E →W ⊗OB →
(
E⊥
)∗ → 0,
0→ E⊥ →W ∗ ⊗OB → E∗ → 0.
Under this general setting, we have the following natural morphisms:
P(E) P
(
E⊥
)
P(W ) B P(W ∗).
f
}} !! }}
g
!!
Lemma B.1. Let Eb and E⊥b be the fibers over b ∈ B of E and E⊥, respectively. Then it holds
dimP
(
Eb ∩ L⊥r
)
= dimP
(
E⊥b ∩ Lr
)
for any r-dimensional linear subspace Lr ⊂W ∗ and the orthogonal linear subspace L⊥r in W .
Proof. We calculate the dimensions as follows: dim
(
Eb ∩ L⊥r
)
= dim Eb + dimL⊥r − dim
(
Eb +
L⊥r
)
= r + (N − r)− dim
(
Eb + L⊥r
)
= dim
(
E⊥b ∩ Lr
)
. �
The complete intersections X1, X2 in Appendix A.2 may be described, respectively, in general
terms as
XL⊥r
= f−1
(
L⊥r
)
∩ P(E), YLr = g−1(Lr) ∩ P
(
E⊥
)
for a fixed subspace Lr ⊂ W ∗, which we call orthogonal linear sections. Consider the Grass-
mannian G = Gr(r,N) of r-spaces in W ∗ and define the following family of orthogonal linear
sections:
Xr :=
{
([Lr], x) ∈ G× P(E) | f(x) ∈ P
(
L⊥r
)}
,
Yr :=
{
([Lr], y) ∈ G× P(E⊥) | g(y) ∈ P(Lr)
}
.
Also we define
Σ0 :=
{
([Lr], b) ∈ G×B | Eb ∩ L⊥r 6= 0
}
=
{
([Lr], b) ∈ G×B | E⊥b ∩ Lr 6= 0
}
,
Movable vs Monodromy Nilpotent Cones of Calabi–Yau Manifolds 35
where the second equality is valid due to Lemma B.1. Then Xr, Yr are orthogonal linear sections
fibered over Σ0 and, with natural morphisms, they can be arranged in the following diagram:
Xr ×Σ0 Yr
Xr Yr
Σ0
P(E) P
(
E⊥
)
.
⊂ G×B
�� ��
!! }}
oo // (B.1)
Let us introduce the following divisors related to the diagram:
HE := OP(E)(1), HE⊥ := OP(E⊥)(1), HG := OG(1).
Proposition B.2. Abbreviating the pull-back symbols by the morphisms in the diagram (B.1),
we have
KXr×Σ0
Yr = −(N − 2)HG −HE −HE⊥ +KB + 2 det E∗
and
KXr = −(N − 1)HG +KB + det E∗,
KYr = −(N − 1)HG +KB + det
(
E⊥
)∗
.
Proof. We leave the proofs for readers. �
It is easy to recognize that the proofs of the above proposition rely on the projective geometry
behind the diagram (B.1). We will report the proofs elsewhere with some additional properties
which we can extract from the diagram (B.1); for example, we can show that the morphisms
Xr → Σ0, Yr → Σ0 are flopping contractions and the naturally induced birational map Xr 99K Yr
in the diagram is the flop for these contractions.
Proposition B.3. Pushing forward KXr×Σ0
Yr to Xr, and equating toKXr , we have a relation
HE +HE⊥ = det E∗ +HG (B.2)
on Xr. Similarly, we have a corresponding relation on Yr,
HE +HE⊥ = det
(
E⊥
)∗
+HG.
Now restricting the relation (B.2) on Xr to Xr|[Lr]×P(E) = XL⊥r
, we obtain
HE |X
L⊥r
+HE⊥ |X⊥Lr
= det E∗, (B.3)
which is the relation we used in (A.2).
Acknowledgements
The results of this work have been reported by the first named author (S.H.) in several work-
shops; “Modular forms in string theory” at BIRS (2016), “Categorical and Analytic invari-
ants IV” at Kavli IPMU (2016), “Workshop on mirror symmetry and related topics” at Kyoto
University (2016) and “The 99th Encounter between Mathematicians and Theoretical Physi-
cists” at IRMA, Strasbourg (2017). He would like to thank the organizers for the invitations
for the workshops where he had valuable discussions with the participants. Writing this pa-
per started when S.H. was staying at Brandeis University and Harvard University in March,
2017. He would like to thank B. Lian and S.-T. Yau for their kind hospitality and also valuable
discussions during his stay. The authors would like to thank anonymous referees for valuable
comments which helped them improve this paper. This work is supported in part by Grant-in
Aid Scientific Research JSPS (C 16K05105, JP17H06127 S.H. and C 16K05090 H.T.).
36 S. Hosono and H. Takagi
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https://doi.org/10.1016/0550-3213(96)00434-8
https://arxiv.org/abs/hep-th/9606040
https://arxiv.org/abs/1704.00164
1 Introduction
2 Classical mirror symmetry
2.1 Mirror symmetry of Calabi–Yau threefolds
2.1.1 A-structure of X
2.1.2 B-Structure of X*
2.1.3 Mirror symmetry
2.2 Birational geometry and mirror symmetry
2.2.1 Movable cones of X
2.2.2 Compactification of the moduli space MX*cpx
3 Complete intersection Calabi–Yau spaces from Gorenstein cones
3.1 Cones for complete intersections and Calabi–Yau manifolds
3.2 Movable cone of X:=X1
3.3 Mirror symmetry of X
4 Gluing monodromy nilpotent cones I
4.1 B-structures of X*
4.1.1 B-structure at o1
4.1.2 B-structures at o2, o3
4.2 Gluing the monodromy nilpotent cones
4.2.1 Path po2o1
4.2.2 The isomorphisms bo'b0, bo''bo' and bobo''
4.2.3 Groupoid actions on the nilpotent cones
4.2.4 Monodromy relations
4.2.5 Gluing nilpotent cones
4.3 Flopping curves and TE1
4.4 Prepotentials
5 Gluing monodromy nilpotent cones II
5.1 Birational automorphisms of infinite order
5.2 Mirror family of X
5.3 B-structure of X at the origin o
5.4 Gluing the monodromy nilpotent cone o
5.4.1 Path poEio, i=1,2
5.4.2 Monodromy around Ei
5.4.3 Monodromy relations
5.4.4 Gluing nilpotent cones
5.5 Flopping curves and TE1
6 Summary and discussions
A Proof of Lemmas 3.5 and 3.6
A.1 Proof of Lemma 3.5
A.2 Proof of Lemma 3.6
B Linear duality
References
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| id | nasplib_isofts_kiev_ua-123456789-209533 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:06:20Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Hosono, S. Takagi, H. 2025-11-24T10:44:00Z 2018 Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds / S. Hosono, H. Takagi // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 46 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14E05; 14E07; 14J33; 14N33 arXiv: 1707.08728 https://nasplib.isofts.kiev.ua/handle/123456789/209533 https://doi.org/10.3842/SIGMA.2018.039 We study mirror symmetry of complete intersection Calabi-Yau manifolds that have birational automorphisms of infinite order. We observe that movable cones in birational geometry are transformed, under mirror symmetry, to the monodromy nilpotent cones, which are naturally glued together. The results of this work have been reported by the first named author (S.H.) in several workshops; “Modular forms in string theory” at BIRS (2016), “Categorical and Analytic invariants IV” at Kavli IPMU (2016), “Workshop on mirror symmetry and related topics” at Kyoto University (2016) and “The 99th Encounter between Mathematicians and Theoretical Physicists” at IRMA, Strasbourg (2017). He would like to thank the organizers for the invitations to the workshops where he had valuable discussions with the participants. Writing this paper started when S.H. was staying at Brandeis University and Harvard University in March 2017. He would like to thank B. Lian and S.-T. Yau for their kind hospitality and also valuable discussions during his stay. The authors would like to thank anonymous referees for their valuable comments, which helped them improve this paper. This work is supported in part by Grant-in-Aid Scientific Research JSPS (C 16K05105, JP17H06127 S.H., and C 16K05090 H. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds Article published earlier |
| spellingShingle | Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds Hosono, S. Takagi, H. |
| title | Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds |
| title_full | Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds |
| title_fullStr | Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds |
| title_full_unstemmed | Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds |
| title_short | Movable vs Monodromy Nilpotent Cones of Calabi-Yau Manifolds |
| title_sort | movable vs monodromy nilpotent cones of calabi-yau manifolds |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209533 |
| work_keys_str_mv | AT hosonos movablevsmonodromynilpotentconesofcalabiyaumanifolds AT takagih movablevsmonodromynilpotentconesofcalabiyaumanifolds |