Hodge Numbers from Picard-Fuchs Equations
Given a variation of Hodge structure over P¹ with Hodge numbers (1,1,…,1), we show how to compute the degrees of the Deligne extension of its Hodge bundles, following Eskin-Kontsevich-Möller-Zorich, by using the local exponents of the corresponding Picard-Fuchs equation. This allows us to compute th...
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| Cite this: | Hodge Numbers from Picard-Fuchs Equations / C.F. Doran, A. Harder, A. Thompson // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 31 назв. — англ. |
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| citation_txt | Hodge Numbers from Picard-Fuchs Equations / C.F. Doran, A. Harder, A. Thompson // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 31 назв. — англ. |
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| description | Given a variation of Hodge structure over P¹ with Hodge numbers (1,1,…,1), we show how to compute the degrees of the Deligne extension of its Hodge bundles, following Eskin-Kontsevich-Möller-Zorich, by using the local exponents of the corresponding Picard-Fuchs equation. This allows us to compute the Hodge numbers of Zucker's Hodge structure on the corresponding parabolic cohomology groups. We also apply this to families of elliptic curves, K3 surfaces and Calabi-Yau threefolds.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 045, 23 pages
Hodge Numbers from Picard–Fuchs Equations
Charles F. DORAN †1, Andrew HARDER †2 and Alan THOMPSON †3†4
†1 Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta,
Edmonton, AB, T6G 2G1, Canada
E-mail: charles.doran@ualberta.ca
†2 Department of Mathematics, University of Miami, 1365 Memorial Drive, Ungar 515,
Coral Gables, FL, 33146, USA
E-mail: a.harder@math.miami.edu
†3 Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, UK
E-mail: a.thompson.8@warwick.ac.uk
†4 DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road,
Cambridge, CB3 0WB, UK
Received January 20, 2017, in final form June 12, 2017; Published online June 18, 2017
https://doi.org/10.3842/SIGMA.2017.045
Abstract. Given a variation of Hodge structure over P1 with Hodge numbers (1, 1, . . . , 1),
we show how to compute the degrees of the Deligne extension of its Hodge bundles, following
Eskin–Kontsevich–Möller–Zorich, by using the local exponents of the corresponding Picard–
Fuchs equation. This allows us to compute the Hodge numbers of Zucker’s Hodge structure
on the corresponding parabolic cohomology groups. We also apply this to families of elliptic
curves, K3 surfaces and Calabi–Yau threefolds.
Key words: variation of Hodge structures; Calabi–Yau manifolds
2010 Mathematics Subject Classification: 14D07; 14D05; 14J32
1 Introduction
The goal of this paper is to compute the Hodge numbers of the parabolic cohomology groups of
a variation of Hodge structure (VHS) in the case where we know the Picard–Fuchs equation.
In more detail, let us assume that C is a smooth quasiprojective curve which bears an R-VHS
of weight k. If V is the underlying local system and j : C ↪→ C is the embedding of C into its
smooth completion, then Zucker [31] showed that H1(C, j∗V) (which we will call the parabolic
cohomology of V) bears a pure Hodge structure of weight k + 1. Moreover, Zucker also showed
that this Hodge structure is closely related to the filtrands of the Leray spectral sequence when
our VHS comes from a family of manifolds, so one can use the Hodge structure on the parabolic
cohomology groups to understand the Hodge structure of fibrations and vice versa.
More recently, following work of Morrison and Walcher [26, 30], interest has arisen in deter-
mining when the Hodge structure on the parabolic cohomology of specific families of Calabi–Yau
threefolds admits rational (2, 2) classes, since such classes correspond to possible normal func-
tions. Morrison and Walcher [26] gave an exciting interpretation of such normal functions of
families of Calabi–Yau threefolds in terms of D-branes.
Following this, del Angel, Müller-Stach, van Straten, and Zuo [3] sought further examples of
such normal functions, by studying Hodge structures on the parabolic cohomology of families of
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:charles.doran@ualberta.ca
mailto:a.harder@math.miami.edu
mailto:a.thompson.8@warwick.ac.uk
https://doi.org/10.3842/SIGMA.2017.045
http://www.emis.de/journals/SIGMA/modular-forms.html
2 C.F. Doran, A. Harder and A. Thompson
Calabi–Yau threefolds whose underlying VHS’s are pull-backs of VHS’s on the thrice-punctured
sphere with b3 = 4; such VHS’s with b3 = 4 were classified by Doran and Morgan [12], who
found 14 cases. They reduced these computations to the computation of the degree of the quasi-
canonical extension of the Hodge bundles of these VHS’s. However, it appears as if the authors
of [3] were not able to compute these degrees in great generality, thus many entries in their
tables are left blank. Further progress was made by Holborn and Müller-Stach [23], however it
appears that they were still not able to complete the table from [3]. Some related computations
of degrees of quasi-canonical extensions of Hodge bundles have also been carried out by Green,
Griffiths, and Kerr [17].
Recently, a preprint of Eskin, Kontsevich, Möller, and Zorich [13, Section 6] suggested a way
to compute the degrees of Hodge bundles of R-VHS’s of (1, 1, . . . , 1)-type, when the Picard–
Fuchs operators controlling these R-VHS’s are known. The goal of this paper is to present the
technique of [13] in greater generality1 and apply it to an array of situations. In particular, we
offer a completion of the tables of [3].
The application most germane to the previous work of the authors is to the study of threefolds
fibred by lattice polarized K3 surfaces. In [7, 8, 9, 10], we constructed Calabi–Yau threefolds
by first choosing convenient families of lattice polarized K3 surfaces X → P1, then performing
base change along suitable maps g : P1 → P1. One of the main results of this work [10] is
a classification of all Calabi–Yau threefolds fibred by Mn-polarized K3 surfaces, where Mn
denotes the rank 19 lattice Mn := E8 ⊕ E8 ⊕ H ⊕ 〈−2n〉. In order to restrict the number of
cases that we needed to check geometrically, the results and techniques of this paper helped
immensely.
In more generality, our results allow us to compute the Hodge numbers of the parabolic
cohomology H1(P1, j∗V), where X → P1 is any smooth projective threefold fibred by K3 surfaces
over P1 with generically Picard rank 19 fibres, and V is the corresponding local system of
cohomology over the complement of the critical values of the fibration. By the Leray spectral
sequence [31, Section 15], this is a direct summand of the Hodge structure on H3(X ,Q), and
its complement has no (3, 0) or (0, 3) part, so in fact this computes h3,0(X ) and places a lower
bound on h2,1(X ).
Finally, we note that in the case of an elliptic fibration, the degree of the Hodge bundle can
be computed geometrically by putting the fibration into Weierstrass normal form
Y 2 = X3 + g2X + g3
and computing the degrees of g2 and g3. In the case of K3 surface fibrations (and to an even
greater extent for Calabi–Yau threefold fibrations) we do not have such a normal form available,
because there are many families of K3 surfaces polarized by rank 19 lattices over P1. One can
construct normal forms corresponding to each polarizing lattice, as we have done in [9, 10]
for some specifically chosen lattices, but this approach quickly becomes intractable: given an
arbitrary rank 19 lattice L, it is very difficult to find explicit representatives for all L-polarized
K3 surfaces (see [5] for some discussion of this problem). The work in this paper allows us to
bypass such considerations in the presence of a known Picard–Fuchs equation. In a sense this is
antipodal to the work of Fujino [16], who computes the Hodge bundles of K3 surface fibrations
with only unipotent monodromy.
1.1 Structure of this paper
Section 2 begins with a brief discussion of the necessary background in Hodge theory and the
theory of ordinary differential equations, following [13], which we require in order to state the
1As stated, the results of [13] apply only to VHS’s of (1, 1, 1, 1)-type, though it is clear, even in their exposition,
that their approach is more general.
Hodge Numbers from Picard–Fuchs Equations 3
main results in Section 2.4. Section 2 concludes with a discussion of how these results are
affected by base change.
The remainder of the paper presents some applications of this theory. Firstly, in Section 3,
we apply the results of Section 2 to the case of elliptic fibrations. Here we show how to compute
the Hodge numbers of an elliptic surface over P1 directly from its Picard–Fuchs equation.
In Section 4, we move on to analyze (1, 1, 1)-type R-VHS’s over P1, which arise in the context
of K3 fibrations. We again compute their Hodge numbers using the corresponding Picard–Fuchs
equations, proving a Hodge bundle formula for such fibrations. In particular, we can use this to
place constraints on the possible K3 fibrations that can arise on a Calabi–Yau threefold, which
allows us to recover an approximate form of a result from [7].
Finally, in Section 5, we will consider the case of Calabi–Yau threefold fibrations over P1.
Here we achieve our main goal and complete the computations of [3] and [23]. Finally, we
conclude with a result that allows us to constrain the possible Calabi–Yau fourfolds which may
admit fibrations by the quintic mirror Calabi–Yau threefold. The results we use to do this apply
in greater generality; we expect similar results to hold for any Calabi–Yau threefolds X with
(1, 1, 1, 1)-type Hodge structure on H3(X,Q).
2 Background and important results
In this section we will develop the necessary background material and state a number of results
that are necessary for the subsequent computations in Sections 3, 4 and 5.
2.1 Variations of Hodge structure
Let us begin with a real variation of polarizable Hodge structure over a quasi-projective curve C.
We denote this R-VHS by (V,F•,∇), where V is the underlying real local system, F• is the Hodge
filtration on V := V ⊗ OC and ∇ is the Gauss–Manin connection. We require that ∇ satisfy
Griffiths transversality [18, 19, 22]; in other words, ∇(Fi) should be contained in Fi+1 ⊗ ΩC .
Following Deligne [4], we can canonically extend the bundle V to a bundle on the smooth
completion C of C, as follows. Near a point p ∈ ∆ = C \ C, we may choose a chart so that p
is the center of a disc D. Let V0 be a fibre of V near p and suppose that monodromy around p
acts on V0 by a quasi-unipotent transformation T . Define subspaces Wα of V0 by
Wα :=
{
v ∈ V0 : (T − ζα)kv = 0
}
,
where ζα = e2πiα and α is chosen to be in the interval [0, 1). These vector spaces are zero for all
but finitely many values of α ∈ [0, 1), and we have a direct sum decomposition V0 =
⊕
αWα. The
vector spaces Wα over the different points of the punctured disc D\{p} define a sub-bundle Wα
of V.
Let Tα denote the action of monodromy on Wα; note that ζ−1
α Tα is unipotent by construction.
Define
Nα :=
−1
2πi
log(Tα),
where the branch of the logarithm is chosen so that the unique eigenvalue of Nα is in the
interval [0, 1). Then for any section v of e∗Wα, where e(z) := exp(2πiz) denotes the complex
exponential, we may define
ṽ(z) = exp(2πiz(Nα − α))v(z),
where z ∈ H is an element of the upper half-plane. Since v(z + 1) = T (v(z)), we see that ṽ(z)
is invariant under the translation z 7→ z + 1, so ṽ defines a holomorphic section of V on the
4 C.F. Doran, A. Harder and A. Thompson
punctured discD∗. Deligne’s canonical extension of V is the bundle onD which is the OD-module
spanned by all such vectors over all α ∈ [0, 1). This is denoted V.
The Hodge filtration F• extends to a Hodge filtration on V, which we denote by F
•
. The
graded pieces of F
•
will be denoted Ep,`−p = F
p
/F
p−1
, where ` is the length of the Hodge
filtration.
Following [13, Section 2], there is a parabolic filtration of the fibre Vp of V over each point
p ∈ ∆, given by V ≥β :=
⊕
α≥β Vα, where Vα is the subspace of Vp spanned by local sections of V
coming from sections of e∗Wα. The parabolic degree of V is given by
degpar V := degV +
∑
p∈∆
α∈[0,1)
α dimVα. (2.1)
This parabolic filtration on the fibres of V over points in ∆ extends to a parabolic filtration
on the fibres of Ep,`−p; we denote its graded pieces by E
p,`−p
α . The connection ∇ extends to
a connection on V with logarithmic poles along ∆ or, in other words,
∇ : V −→ V⊗ ΩC(∆).
This connection is horizontal with respect to the Hodge filtration F
•
, so we obtain morphisms
of bundles
θi−1 : E`−i,i −→ E`−i−1,i+1 ⊗ ΩC(∆).
This map also respects the local filtration at each point p ∈ ∆, i.e.,
∇ : V ≥βp −→ V ≥βp ⊗ ΩC(∆).
The following useful result appears as part of the proof of [13, Theorem 6.1].
Lemma 2.1. Assume that C = P1. Then
degpar E
p,q = −degpar E
q,p.
The following two examples will be useful in later portions of this paper.
Example 2.2 (elliptic curves). Let us take a degeneration of elliptic curves over the unit disc
in C, with central fibre of Kodaira type IV∗. In this case, one can check (see [25, Table VI.2.1])
that the local monodromy action on the first integral cohomology of a general fibre has matrix(
−1 −1
1 0
)
.
This matrix has eigenvalues equal to e2πi/3 and e4πi/3. Therefore, the spaces V1/3 and V2/3
are both 1-dimensional. We can ask how these spaces interact with the canonical extension
of the Hodge filtration. We know that the map ∇ : F
0 → F
1 ⊗ ΩD(0) respects the filtration
C2 = V ≥1/3 ⊃ V ≥2/3 = C and thus the image of ∇ in F
1⊗ΩD(0) must be V ≥2/3 = V2/3, since ∇
is not zero. Therefore E
0,1
2/3 = E0,1 and E
0,1
1/3 = 0. Similar arguments show that E
1,0
1/3 = E1,0 and
E
1,0
2/3 = 0.
Using Kodaira’s classification of germs of one parameter degenerations of elliptic curves,
one can compute parabolic filtrations on the Hodge bundles corresponding to all non-unipotent
degenerations of elliptic curves in the same way. The result is given in Table 1, which lists the
values of α such that E
1,0
α = E1,0 and E
0,1
α = E0,1 for each type of singular fibre.
Using this, the parabolic degrees of E1,0 and E0,1 can be determined easily once one knows their
degrees. Furthermore, for an elliptic fibration over P1, we may combine this with equation (2.1)
and Lemma 2.1 to show that −degE1,0−degE0,1 is equal to the number of fibres around which
monodromy is not unipotent. This is precisely [23, Remark 2.4].
Hodge Numbers from Picard–Fuchs Equations 5
Fibre type E1,0 E0,1
In 0 0
I∗n 1/2 1/2
II or II∗ 1/6 5/6
III or III∗ 1/4 3/4
IV or IV∗ 1/3 2/3
Table 1. Values of α such that Ep,qα = Ep,q for degenerations of elliptic curves.
Monodromy matrix E2,0 E1,1 E0,2
Tun 1/2 1/2 1/2
T 2
un 0 0 0
Ti 1/4 1/2 3/4
−Ti 1/4 0 3/4
T 2
i 1/2 0 1/2
Tω 1/6 1/2 5/6
T 2
ω 1/3 0 2/3
Tnod 0 1/2 0
Table 2. Values of α such that Ep,qα = Ep,q for (1, 1, 1)-VHS’s.
Example 2.3 ((1, 1, 1) variations of Hodge structure). Let us take a degeneration X of K3
surfaces over the complex unit disc D, so that the restriction of X to D∗ := D \ {0} is an M -
polarized family of K3 surfaces, in the sense of [7, Definition 2.1], for some rank 19 lattice M .
The transcendental lattices of the general fibres give rise to a VHS of type (1, 1, 1) over D∗.
One can generally describe all possible monodromy matrices that can underlie such a (1, 1, 1)-
VHS. Up to sign, they are just symmetric squares of parabolic and elliptic elements of SL2(R);
this comes from the fact that O(2, 1) is, up to sign, a symmetric square of the group SL2(R).
By [23, Lemma 3.1], the monodromy of X around 0 ∈ D can be written in Jordan canonical
form as eitherλ 1 0
0 λ 1
0 0 λ
or
λ1 0 0
0 λ2 0
0 0 λ3
for some λ, λi ∈ C roots of unity with
3∑
i=1
λi ∈ Z. Cases that are relevant to our work [10] are
Tun =
−1 1 0
0 −1 1
0 0 −1
, Ti =
√−1 0 0
0 −
√
−1 0
0 0 −1
,
Tω =
−e2πi/3 0 0
0 −e4πi/3 0
0 0 −1
, Tnod =
−1 0 0
0 1 0
0 0 1
,
and their powers.
As in the previous example, we can compute the filtration on the degenerate Hodge structure
from these matrices. The result is given in Table 2, which lists the values of α such that
E
2,0
α = E2,0, E1,1
α = E1,1, and E
0,2
α = E0,2 for each type of VHS.
For a fibration of K3 surfaces over P1, we may combine this with equation (2.1) and Lemma 2.1
to show that −degE2,0 − degE0,2 computes the number of fibres around which we have mono-
dromy matrix with Jordan normal form Tun, Ti, −Ti, T 2
i , Tω or T 2
ω . If we adjust the definition
6 C.F. Doran, A. Harder and A. Thompson
of II in [23, Section 3] to be the set of points in ∆ whose monodromy matrices have at least two
eigenvalues not equal to 1, then [23, Remark 3.4] holds.
2.2 Picard–Fuchs equation of a variation of Hodge structure
Now let us take a look at how the Picard–Fuchs equation of a variation of Hodge structure
arises; more details for everything in this section and the next may be found in [13, Section 6.4].
We start with the same basic setup as before: a variation of Hodge structure (V,F•,∇) over
a quasi-projective curve C with its canonical extension to C. For the sake of simplicity, we will
assume that C = P1.
If we now choose a global section2 σ of the Serre twist F
`
(i), where ` is the length of the
Hodge filtration and i is some integer, then we can build a differential equation associated to σ.
Taking a polarization 〈−,−〉 on V, we can assign to σ a set of multivalued functions
sv(t) = 〈σ, v〉,
where v is some flat local section on V (or any flat local section v of V := V ⊗ OC). The
functions sv(t) give multivalued meromorphic functions on P1, with monodromy occurring only
around points in ∆ = C \ C. One can then produce a global differential equation Lσ, of rank
at most the rank of V on C, whose solution set is the set of functions sv(t). The differential
operator Lσ is a Fuchsian ODE (which will be discussed in the following section), called the
Picard–Fuchs equation.
The equation Lσ is defined as follows. For an appropriate choice of affine coordinate t on
C = P1, we may assume that ∞ ∈ ∆, and therefore that the vector field d/dt is a global vector
field on C = P1 \∆. If α and β are sections of V, then we have
d
dt
〈α, β〉 = 〈∇d/dtα, β〉 ± 〈α,∇d/dtβ〉,
where ∇d/dtα is defined to be ∇(α) ∈ V⊗ ΩC paired with d/dt. Therefore,
dsv(t)
dt
= 〈∇d/dtσ, v〉,
since v is chosen to be a flat local section of V.
Since ∇id/dt(σ) all live in V, if r = rankV, then σ,∇d/dtσ, . . . ,∇rd/dtσ must be linearly depen-
dent over the field of meromorphic functions on C. Therefore, we can write an equation
∇nd/dtσ + f1(t)∇n−1
d/dtσ + · · ·+ fn(t)σ = 0
for some minimal n ≤ r. Therefore, for any flat local section v of V, we obtain the equation
dnsv(t)
dtn
+ f1(t)
dn−1sv(t)
dtn−1
+ · · ·+ fn(t)sv(t) = 0,
by linearity of the operator 〈−, v〉. In other words, sv(t) is annihilated by the differential operator
Lσ =
dn
dtn
+ f1(t)
dn−1
dtn−1
+ · · ·+ fn(t).
On the other hand, to a rank r ordinary differential equation L with regular singularities, we
can assign to L its solution sheaf Sol(L), which is a subsheaf of OC spanned over C by the local
solutions of L. The solution sheaf of Lσ is precisely the subsheaf of OC spanned locally by the
functions sv(t).
2More generally, if such a section does not exist, one may instead construct a Picard–Fuchs D-module; however
this will not be relevant to our work.
Hodge Numbers from Picard–Fuchs Equations 7
2.3 Fuchsian ODE’s
Let L be a linear ordinary differential equation in one variable, written as
dn
dtn
+
n∑
i=1
fi(t)
(
d
dt
)n−i
,
where the fi are meromorphic functions in t. This equation can be rewritten in terms of the
operator δt = t ddt as
δnt +
n∑
i
gi(t)δ
n−i
t (2.2)
for some meromorphic functions g1, . . . , gn. Indeed, one can check that
tn
(
d
dt
)n
= δt(δt − 1) · · · (δt − n− 1),
so it is easy to translate between the two expressions for L.
A point a ∈ P1 is called nonsingular if fi(t) is holomorphic at a, and is a regular singular
point if (t− a)ifi(t) is holomorphic at a, for all i ∈ {1, . . . , n}. Note that 0 is a regular singular
point if and only if the coefficients gi(t) in equation (2.2) are all holomorphic at 0. A linear
differential equation of one variable with only regular singularities at all points in P1 is called
a Fuchsian ODE. Regular singular points can be divided into apparent singularities, around
which monodromy acts as the identity on the solution sheaf, and actual singularities, where it
does not.
Let L be a Fuchsian ODE with a regular singular point at 0 ∈ P1. The polynomial
Tn +
n∑
i=1
gi(0)Tn−i
is called the indicial equation of L at 0, and its roots are called the characteristic exponents of L
at 0. For an arbitrary point a ∈ P1, one may compute the indicial equation and characteristic
exponents of L at a by making the variable change t 7→ t− a. We will denote the characteristic
exponents of L at a by µa1, . . . , µ
a
n; throughout the paper we will assume that the µai are ordered
by magnitude, so that µa1 ≤ µa2 ≤ · · · ≤ µan. Often, the data of the regular singular points ai and
their characteristic exponents µ
aj
i are arranged into a matrix known as the Riemann scheme
of L, which looks like
a1 a2 · · · ak
µa11 µa21 · · · µak1
...
...
. . .
...
µa1n µa2n · · · µakn
.
Information about the monodromy of the solution sheaf of L can be read off from the Riemann
scheme. For instance, if a is a regular singular point whose characteristic exponents are all
integers, then monodromy around a is unipotent, i.e., all of its eigenvalues are 1. As a partial
converse, if a is an apparent singularity, then the characteristic exponents at a are all integers.
Finally, note that it also makes sense to compute the characteristic exponents at a nonsingular
point a of L. If one does so, one obtains (µa1, µ
a
2, . . . , µ
a
n) = (j, j + 1, . . . , j + n − 1), for some
integer j.
8 C.F. Doran, A. Harder and A. Thompson
Remark 2.4. It is often natural to scale the solutions to a differential equation by a meromor-
phic function h(t) on P1 to obtain a twisted differential equation, denoted hL. This operation
acts on the characteristic exponents of L by an overall scaling hµai = µai +orda(h), where orda(h)
is the order of h at a. Importantly, the differences between the characteristic exponents are not
affected by the process of twisting. In the main results of this paper, we will only need to know
the differences between the characteristic exponents, so knowing Lσ up to twist is enough.
This is important in applications, since one often only knows the Picard–Fuchs equation of
a variation of Hodge structure up to twist: for instance, this is all that the Griffiths–Dwork
method [20, 21] for computing Picard–Fuchs equations of hypersurfaces guarantees. In Theo-
rem 2.7, only the difference in characteristic exponents is relevant; therefore, our Hodge bundle
computations can be done with hL for any meromorphic function h.
Remark 2.5. One can also twist by multivalued functions h on P1. A particular choice of
(possibly) multivalued function h can be made so that hL has fn−1(t) = 0. This is called the
projective normal form of L. The process of twisting by a multivalued function does affect the
differences of characteristic exponents in a significant way. For this reason, we will avoid using
the projective normal form in this paper.
Remark 2.6. The characteristic exponents of L at a point a allow us to compute local solutions
for L near a. We remark that in general, the solutions of Lf = 0 around the point a have the
form tµ
a
i Pi(t) for some holomorphic functions Pi(t), modulo terms with logarithmic singularities
at a. For instance, if all µai ’s are the same, then there is an ordered basis of local solutions {si(t)}
around a given inductively by
si(t) = tµ
a
i Pi(t) +
i−1∑
j=1
sj(t) log t
j!
,
where the Pi are holomorphic functions which do not vanish at a.
If none of the differences between characteristic exponents are integers, then there are no
terms with logarithmic singularities and the local solutions around a all have the form tµ
a
i Pi(t),
for some holomorphic functions Pi(t). In this case, if L is the Picard–Fuchs equation of a variation
of Hodge structure (V,F•,∇), then Deligne’s canonical extension V of V is spanned by the set of
local sections {tbµai cPi(t)} and the subspaces spanned by tbµ
a
i cPi(t) are the parabolic filtrands Vα,
where α := µai −bµai c. A more in-depth analysis of the Frobenius method shows that this is true
in general.
2.4 Important results
Here we state some important results which, in combination, will allow us to compute Hodge
numbers of a variation of Hodge structure of (1, 1, . . . , 1)-type if we know the Picard–Fuchs
equation Lσ. The first is a result of Eskin, Kontsevich, Möller, and Zorich [13], which says that
the characteristic exponents of the Picard–Fuchs equation Lσ corresponding to a variation of
Hodge structure of (1, 1, . . . , 1)-type can be used to compute the Kodaira–Spencer maps on the
bundles Ep,q.
Theorem 2.7 ([13, Lemma 6.3]). Let (V,F•,∇) be a polarized R-VHS of (1, 1, . . . , 1)-type over
a Zariski open subset P1 \∆, let F
•
be its quasi-canonical extension to P1, let Ep,q be the graded
pieces of F
•
, and let Lσ be the Picard–Fuchs differential equation associated to a meromorphic
section of σ of E`,0. Then for i < `/2, the Kodaira–Spencer map θi : E
`−i,i → E`−1−i,1+i⊗ΩP1(∆)
is described as follows.
1. If p is a nonsingular point of Lσ, then θi is a local isomorphism at p.
Hodge Numbers from Picard–Fuchs Equations 9
2. If p is an apparent singularity of Lσ, then θi has cokernel of length µpi+2 − µ
p
i+1 − 1.
3. If p is an actual singularity of Lσ, then θi has cokernel of length bµpi+2c − bµ
p
i+1c.
Remark 2.8. We record the observation that Eskin, Kontsevich, Möller, and Zorich use Theo-
rem 2.7 to compute the parabolic degrees of the Hodge bundles of hypergeometric variations
of Hodge structure. Specifically, they are interested in the 14 VHS’s of (1, 1, 1, 1)-type on P1
found by Doran and Morgan [12]. However, it seems as if the authors of [13] were unaware of
the fact that this result follows easily from the data in [23], in which the degrees of E3,0 and E2,1
are computed for all fourteen (1, 1, 1, 1)-type variations of Hodge structure of hypergeometric
type. We will see later that Theorem 2.7 can be used to perform computations that neither the
authors of [3] nor [23] were able to perform.
This result is useful because, if we know the parabolic filtration of each bundle Ep,q and the
Picard–Fuchs equation Lσ, then knowing the degrees of all maps θi allows us to easily compute
the degrees of Ep,q when each Ep,q is a line bundle (i.e. when Lσ underlies a variation of Hodge
structure of weight d and type (1, 1, . . . , 1)).
Next we will state several results due to del Angel, Müller-Stach, van Straten and Zuo, and
Hollborn and Müller-Stach, which appear in [3, 23]. These results say that, in the situations
that interest us, we can compute Hodge numbers of H1(P1, j∗V) (where j : C ↪→ C = P1 denotes
the inclusion) using the degrees of Ep,q and the local monodromy data of V. According to
Theorem 2.7, all of this information can be obtained from Lσ.
Theorem 2.9 ([3, 23]). Let V be a local system on P1 \∆ which supports a variation of Hodge
structure of weight n and (1, 1, . . . , 1)-type. Then the Hodge number h0,n+1 of H1(P1, j∗V) is
given by
h0,n+1 = h1
(
P1,E0,n
)
= h0
(
P1,OP1
(
−2− degE0,n
))
.
In the case where n = 1 or 2 in the above theorem, then we can deduce the rest of the Hodge
numbers of the parabolic cohomology H1(P1, j∗V) by Hodge symmetry and the following result,
which generalizes the classical Riemann–Hurwitz formula (presumably this result is classically
known, but the only reference that we know is [3]).
Proposition 2.10 ([3, Proposition 3.6]). Let V be a local system on a quasi-projective curve C
and let j : C ↪→ C be the smooth completion. Let γq be the local monodromy matrix around
the point q in ∆ := C \ C, acting on Vp for some basepoint p in C, and define R(q) :=
dimVp − dimVγqp , where Vγqp denotes the part of Vp fixed under γq. Then
h1
(
C, j∗V
)
=
∑
q∈∆
R(q) +
(
2g
(
C
)
− 2
)
rankV.
Finally, if V is a VHS of (1, 1, 1, 1)-type, then Hollborn and Müller-Stach show how to compute
the Hodge number h1,3 of the parabolic cohomology H1(P1, j∗V) from the degrees of its Hodge
bundles.
Theorem 2.11 ([23, Theorem 4.3]). If V supports a variation of Hodge structure of type
(1, 1, 1, 1) over a Zariski open subset j : C ↪→ P1, then the Hodge number h1,3 of H1(P1, j∗V) is
given by
h1,3 = −2 + b− a+ |II|+ |III|+ |IV|,
10 C.F. Doran, A. Harder and A. Thompson
where a = deg(E3,0) and b = deg(E2,1). The sets II and III are the subsets of ∆ = P1 \C whose
local monodromy matrices have Jordan normal forms
1 1 0 0
0 1 1 0
0 0 1 1
0 0 0 1
, and
1 1 0 0
0 1 0 0
0 0 1 1
0 0 0 1
respectively. The set IV is the subset of ∆ for which the monodromy matrix is strictly quasi-
unipotent.
2.5 Base change and characteristic exponents
The following instructive computation will be useful to us later. We want to understand how base
change of a Fuchsian ODE affects its Riemann scheme. More precisely, let us take a Fuchsian
ODE L and a map g : P1 → P1. Pulling-back the solution sheaf of L by g yields a local sys-
tem g∗Sol(L) and, in particular, another ODE g∗L, of the same rank as L, whose solution sheaf
is g∗Sol(L). Our goal in this section is to analyze the local exponents of g∗L. Away from the
ramification points of g, the differential equation g∗L will have the same behaviour as L at
the corresponding point. So we can restrict ourselves to studying the local behaviour of L at
ramification points, and this can be done in the power series ring C[[t]]. Therefore, our question
reduces to a question of base change of a Fuchsian ODE along the map sk = t, for a positive
integer k.
The following computation appears for rank 2 ODEs in [6].
Proposition 2.12 ([6, Lemma 3.21]). Base change of order k at a point p ∈ P1 affects the
characteristic exponents of L by multiplying everything by k.
Proof. Assume that w(t) is a local solution to Lw = 0. By the chain rule, we have that
dw
ds
(
sk
)
= ksk−1dw
dt
(
sk
)
,
therefore
δtw
(
sk
)
= sk
dw
dt
(
sk
)
=
s
k
dw
ds
(
sk
)
=
δs
k
w
(
sk
)
.
Thus if w is a solution to the differential equation
δnt +
n∑
i=1
fi(t)δ
n−i
t ,
then w(sk) solves
δns +
n∑
i=1
kifi
(
sk
)
δn−is .
Suppose that the indicial equation of L at 0 is given by
n∏
i=1
(T − µi),
for µi the local exponents of L at 0. Then the indicial equation of the differential equation in
terms of s is written as
n∏
i=1
(T − kµi).
This proves the proposition. �
Hodge Numbers from Picard–Fuchs Equations 11
Corollary 2.13. After base change, the ramification points of g : P1 → P1 become apparent
singularities of g∗L. If p is a point with characteristic exponents all 0, then for any point
q ∈ g−1(p), the characteristic exponents of g∗L at q are all 0.
Combining this with Theorem 2.7, we can say quite a bit about how the Hodge numbers of
a variation of Hodge structure of type (1, 1, . . . , 1) are altered by base change.
Remark 2.14. We will take this opportunity to emphasize that performing base change on
a variation of Hodge structure and base change on a linear differential equation may not in
fact be compatible. In other words, there is a natural variation of Hodge structure on P1
underlying g∗V with appropriate canonical extensions at the points in g−1(∆). The Hodge
bundles Ep,qg of this variation of Hodge structure may not agree with the pull-backs g∗Ep,q of the
Hodge bundles associated to V. However, the monodromy representation and period maps are
correct, so if Lg is the Picard–Fuchs system of the pulled back variation of Hodge structure, then
g∗L = hLg for some meromorphic function h. Thus the differences between the characteristic
exponents of g∗L are the same as for Lg, so we can apply Theorem 2.7 to compute the degrees
of Ep,qg .
The underlying reason behind this incompatibility is that Deligne’s canonical extension does
not commute with base change. Indeed, Deligne’s canonical extension of g∗V chooses a branch
of the logarithm so that the eigenvalues of the maps Nα at each point of g−1(∆) lie in the
interval [0, 1) (see Section 2.1). However base change acts additively on these eigenvalues, so
may take them out of this interval; the pulled-back variation of Hodge structure may thus
correspond to a different choice of extension of g∗V.
3 Families of elliptic curves
In the remainder of this paper we will study some applications of these results in various settings.
We begin with the case of a family of elliptic curves.
Let L be a rank 2 ODE which underlies a real variation of Hodge structure of weight 1. We
will show that if L is the Picard–Fuchs equation of a family of elliptic curves S, then we can
determine the geometry of S from L.
First, we will restate Theorem 2.7 in the case of a (1, 1)-VHS.
Corollary 3.1. Let L be a rank 2 Fuchsian ODE which underlies an R-VHS of weight 1, with
regular singular points at ∆ in P1. Then the map θ : E1,0 → E0,1 ⊗ ΩP1(∆) has the following
structure.
1) if p is a nonsingular point of L, then θ is an isomorphism at p,
2) if L has an apparent singularity at p, then θ has cokernel of length µp2 − µ
p
1 − 1 at p,
3) if p ∈ ∆, then θ has cokernel of length bµp2c − bµ
p
1c.
The following theorem follows easily.
Theorem 3.2. Suppose that L underlies a family of elliptic curves over P1. Let aL denote the
number of points in ∆ at which the local monodromy of L is strictly quasi-unipotent, and let ∆a
denote the set of points at which L has an apparent singularity. Then
degE0,1 =
1
2
(
2− aL +
∑
p∈∆
(
bµp2c − bµ
p
1c − 1
)
+
∑
p∈∆a
(
µp2 − µ
p
1 − 1
))
.
12 C.F. Doran, A. Harder and A. Thompson
Remark 3.3. If we note that the characteristic exponents µpi are integers when p is nonsingular
or an apparent singularity, and that µp2 = µp1 + 1 when p is a non-singular point, then we can
simplify the formula in Theorem 3.2 to
degE0,1 =
1
2
(
2− aL +
∑
p∈P1
(
bµp2c − bµ
p
1c − 1
))
.
However, the form given in Theorem 3.2 is simpler to work with from a computational stand-
point, as both sums are clearly finite.
Proof of Theorem 3.2. By Example 2.2, we see that
degE1,0 + degE0,1 = −aL,
and Corollary 3.1 gives that the map θ has cokernel of length∑
p∈∆a
(
µp2 − µ
p
1 − 1
)
+
∑
p∈∆
(
bµp2c − bµ
p
1c
)
.
Since deg(E0,1 ⊗ ΩP1(∆)) = |∆| − 2 + degE0,1, it follows that
degE0,1 + |∆| − 2 = −degE0,1 − aL +
∑
p∈∆a
(
µp2 − µ
p
1 − 1
)
+
∑
p∈∆
(
bµp2c − bµ
p
1c
)
.
Thus
degE0,1 =
1
2
(
2− aL +
∑
p∈∆a
(
µp2 − µ
p
1 − 1
)
+
∑
p∈∆
(
bµp2c − bµ
p
1c − 1
))
as required. �
The term (µp2 − µ
p
1 − 1) should be thought of as an invariant which counts the ramification
of the j-function at p ∈ ∆a; Doran makes this precise in [6].
Remark 3.4. From this, we may use Theorem 2.9 to compute h0,2 for the parabolic cohomology
of our elliptic fibration. In [2], Cox and Zucker show that this h0,2 computes the dimension of
a particular space of modular forms. The computations in this section can be thought of as
giving a way to compute the dimension of this space of modular forms using the Picard–Fuchs
equation of the elliptic surface.
4 Families of K3 surfaces
Next we turn our attention to families of K3 surfaces, as studied in Example 2.3. Theorem 2.7
can be rephrased in the context of the corresponding (1, 1, 1)-variations of Hodge structure, as
follows. Recall that in this case we have the Kodaira–Spencer maps θ0 : E2,0 → E1,1 ⊗ ΩP1(∆)
and θ1 : E1,1 → E0,2 ⊗ ΩP1(∆).
Corollary 4.1. If L is a Picard Fuchs equation corresponding to a (1, 1, 1)-type variation of
Hodge structure, then
1) if p is a nonsingular point of L, then θ0 is an isomorphism,
2) if p is an apparent singularity of L, then θ0 has cokernel of length µp2 − µ
p
1 − 1 at p,
3) if p is a regular singular point of L, then θ0 has cokernel of length bµp2c − bµ
p
1c.
Hodge Numbers from Picard–Fuchs Equations 13
Let U = P1 \∆ denote a Zariski open set and let j : U ↪→ P1 denote the embedding. Suppose
that X → U is an M -polarized family of K3 surfaces, in the sense of [7, Definition 2.1], for some
rank 19 lattice M . The transcendental lattices of the fibres of X define a VHS V of type (1, 1, 1)
over U ; we assume that all of the monodromy matrices of this VHS take the forms Tx considered
in Example 2.3. Finally, let L denote the Picard–Fuchs equation underlying this (1, 1, 1)-VHS.
By Lemma 2.1, it is easy to see that degpar E
1,1 = 0, so the degree of E1,1 can be computed
easily using equation (2.1): indeed, if a1/2 is the number of points p at which monodromy is
of type Tun, Ti, Tω or Tnod (see Example 2.3), then degE1,1 = −1
2a1/2. We let af denote the
number of points whose monodromy matrices look like Tun, Ti, −Ti, T 2
i , Tω and T 2
ω . Finally,
let ∆a denote the set of points at which L has an apparent singularity.
Proposition 4.2. If L underlies a family of K3 surfaces X → U as above, then we have
deg
(
E0,2
)
= 2 +
1
2
a1/2 − af +
∑
p∈∆a
(
µp2 − µ
p
1 − 1
)
+
∑
p∈∆
(
bµp2c − bµ
p
1c − 1
)
.
Proof. This is proved by a similar calculation to Theorem 3.2. �
Remark 4.3. As in the case of elliptic curves, we may simplify this formula to the shorter, but
less transparent, form
deg
(
E0,2
)
= 2 +
1
2
a1/2 − af +
∑
p∈P1
(
bµp2c − bµ
p
1c − 1
)
.
Using this, one can easily use Theorem 2.9 to compute h0,3 of the Hodge structure on
H1(P1, j∗V).
Corollary 4.4. The Hodge number h0,3 of the Hodge structure on H1(P1, j∗V) is
h0,3 = h0
(
P1,O1
P
(
af − 4− 1
2
a1/2 −
∑
p∈∆a
(
µp2 − µ
p
1 − 1
)
−
∑
p∈∆
(
bµp2c − bµ
p
1c − 1
)))
.
Now we may ask ourselves: when can such a fibration by K3 surfaces have a Calabi–Yau
threefold total space?
Definition 4.5. A weight n Hodge structure with Hodge numbers hp,q is called Calabi–Yau if
hn,0 = 1.
If a variety X is Calabi–Yau of dimension n, then the Hodge structure on Hn(X,Q) is
certainly Calabi–Yau, but the converse is not true. For example, look at any blow-up of an
honest Calabi–Yau variety, or a Kulikov surface.
According to Corollary 4.4, a family of K3 surfaces X → U has Calabi–Yau Hodge structure
on H1(P1, j∗V) if and only if
af = 4 +
1
2
a1/2 +
∑
p∈∆a
(
µp2 − µ
p
1 − 1
)
+
∑
p∈∆
(
bµp2c − bµ
p
1c − 1
)
. (4.1)
By the degeneration of the Leray spectral sequence [31, Section 15], for any compactification X
of X , this parabolic cohomology group is a direct summand of the Hodge structure on H3(X ,Q),
and its complement has no (3, 0) or (0, 3) part (see [10]). Thus, if equation (4.1) does not hold,
then X → P1 has no compactification which is a Calabi–Yau threefold. In [10] we will study
the converse problem in specific examples.
14 C.F. Doran, A. Harder and A. Thompson
4.1 Base change of a hypergeometric family
We illustrate these results with an example regarding families of K3 surfaces and their pullbacks.
Let us take the family of mirror quartic K3 surfaces written as the compactifications in P3 of
the fibres of the Laurent polynomial
f(x, y, z) =
(x+ y + z + 1)4
xyz
.
Narumiya and Shiga [27, Section 5] show that this family has Picard–Fuchs equation which is
hypergeometric and may be written as
δ3 + t
(
δ + 1
4
) (
δ + 1
2
) (
δ + 3
4
)
.
The local monodromies around its singular points are of type T 2
un, Ti, and Tnod, over 0,∞, and 1
respectively, so the Riemann scheme of this differential operator is
0 ∞ 1
0 1/4 0
0 1/2 1/2
0 3/4 1
.
Now consider the base change of this local system along a map g : P1 → P1. Let E
i,j
g denote
the appropriate Hodge bundles of the pulled-back local system g∗V and let hi,jg be the Hodge
numbers of its parabolic cohomology. Let d be the degree of g, and set k, `, m to be the numbers
of points over 0, ∞ and 1 respectively. Let
r =
∑
p∈P1\{0,1,∞}
(ep − 1)
denote the degree of ramification of g away from {0, 1,∞}. We will write [y1, . . . , y`] for the
partition of d encoding the ramification profile over ∞. Finally, define
Dg := af − a1/2 − 4−
∑
p∈∆a
(µp2 − µ
p
1 − 1)−
∑
p∈∆
(
bµp2c − bµ
p
1c − 1
)
,
where all terms in this expression are computed for the pulled-back local system g∗V. Then we
have the following proposition.
Proposition 4.6. With assumptions and notation as above,
Dg = af − 2 +
∑̀
i=1
⌊yi
4
⌋
.
In particular, h0,3
g = h0
(
P1,OP1(Dg)
)
.
Proof. It follows from the definition that af is equal to the number of points over 0 with
ramification of order not divisible by 4. Let `odd and modd denote the number of points over 0
and 1, respectively, where the order of ramification is odd. Then
a1/2 = modd + `odd.
Apparent singularities appear only at ramification points of g away from points over {0, 1,∞},
and each ramification point has µp2 − µ
p
1 − 1 equal to ep − 1. Therefore∑
∆a
(
µp2 − µ
p
1 − 1
)
=
∑
p∈P1\{0,1,∞}
(ep − 1) = r.
Hodge Numbers from Picard–Fuchs Equations 15
We split the set ∆ into three components, corresponding to points over 0, 1, and ∞ respec-
tively, and denoted ∆0, ∆1, and ∆∞. The corresponding sums are
∑
p∈∆0
(
bµp2c − bµ
p
1c − 1
)
=
1
2
(d− `odd)−
∑̀
i=1
(
byi
4
c
)
− `,
∑
p∈∆1
(
bµp2c − bµ
p
1c − 1
)
=
1
2
(d−modd)−m,
∑
p∈∆∞
(
bµp2c − bµ
p
1c − 1
)
= k.
Substituting everything into Dg, we obtain
Dg = af − 4− d− r + k + `+m+
∑̀
i=1
⌊yi
4
⌋
.
Now, since g is a map from P1 to P1, the Riemann–Hurwitz formula gives
2d− 2 =
∑
p
(ep − 1) = r + (d− `) + (d−m) + (d− k),
which implies that k + `+m− d− r = 2. Making this substitution, we find that
Dg = af − 2 +
∑̀
i=1
⌊yi
4
⌋
,
which proves the proposition. �
As a corollary, we can approximately recover [9, Proposition 2.4] (see also [10] for some further
applications of this result).
Corollary 4.7. The Hodge structure of g∗V is Calabi–Yau if and only if one of the following
two conditions holds:
1) the number ` is equal to 2 and 1 ≤ y1, y2 ≤ 4; or
2) the number ` is equal to 1 and 5 ≤ y1 ≤ 8.
Proof. By Corollary 4.4 and Proposition 4.6, we may assume that
af +
∑̀
i=1
byi
4
c = 2.
Therefore, we have at most two points over 0. If we have two points over 0, then they must
both have 1 ≤ yi ≤ 4. If there is a single point over 0, then 5 ≤ y1 ≤ 8. �
5 Families of Calabi–Yau threefolds
In this section, we will first apply the results of Section 2.4 to complete the computations of [3]
and [23]. Then we will determine the Hodge numbers of some specific fourfolds fibred by quintic
mirror Calabi–Yau threefolds.
16 C.F. Doran, A. Harder and A. Thompson
5.1 Inhomogeneous Picard–Fuchs equations and Calabi–Yau threefolds
In this section, we will perform some computations related to work of del Angel, Müller-Stach,
van Straten, and Zuo [3]. These authors were interested in determining situations in which
families of Calabi–Yau threefolds admit normal functions; here a normal function is an algebraic
cycle on each fibre of a family of varieties, which is homologous to zero on all fibres.
In particular, if we have some algebraic (2, 2)-class Z on a family of Calabi–Yau threefolds,
whose restriction to each fibre is homologous to zero, then Z determines a multivalued func-
tion ΦZ on the base of the fibration. From such a normal function, one obtains an inhomogeneous
extension of the Picard–Fuchs equation Lσ which is satisfied by ΦZ . In other words,
LσΦZ = gZ ,
for some holomorphic function gZ . Morrison and Walcher [26, 30] studied a specific such normal
function on the Fermat family of quintics, and showed that ΦZ can be interpreted as the domain
wall tension of certain D-branes.
The computations in [3] aimed to determine whether such normal functions exist on other
1-parameter families of Calabi–Yau threefolds. Evidence for the existence of normal functions
can be found by determining whether the parabolic cohomology of Lσ has classes of type (2, 2)
or not. They looked at the 14 hypergeometric examples of Doran and Morgan [12] and their
pullbacks along maps [t : s] 7→ [td : sd], for certain values of d, in an effort to determine whether,
after such base change, the 14 families could admit interesting normal functions.
In both [3] and the subsequent work of Hollborn and Müller-Stach [23], the authors were not
able to compute the degrees of the bundles E2,1 in many situations, and therefore were not able
to determine the Hodge numbers of the parabolic cohomology groups that they were interested
in. By applying the results of Section 2.4, we are now able to finish their computations.
Example 5.1. Let’s compute the Hodge numbers of the VHS obtained from the second hy-
pergeometric example in the list of Doran and Morgan [12]. This example is obtained in the
following way. Begin with the hypergeometric operator
δ4 − t
(
δ +
1
10
)(
δ +
3
10
)(
δ +
7
10
)(
δ +
9
10
)
.
This operator underlies a (1, 1, 1, 1)-type rational VHS, and has local monodromy matrices
T0 =
1 1 0 0
0 1 1 0
0 0 1 1
0 0 0 1
, T1 =
1 0 0 0
0 1 0 0
0 0 1 1
0 0 0 1
, T∞ =
ζ10 0 0 0
0 ζ3
10 0 0
0 0 ζ7
10 0
0 0 0 ζ9
10
,
where ζ10 is a primitive tenth root of unity. Its Riemann scheme is
0 1 ∞
0 0 1/10
0 1 3/10
0 1 7/10
0 2 9/10
.
Let us take the base change of this differential equation under the map of degree 5 which
is totally ramified over 0 and ∞. The new differential equation has seven singular points and
Hodge Numbers from Picard–Fuchs Equations 17
Riemann scheme
0 t5 − 1 = 0 ∞
0 0 1/2
0 1 3/2
0 1 7/2
0 2 9/2
.
The points over 1 all have monodromy of type T1, the point over 0 has monodromy of type T0,
and the point over ∞ has monodromy matrix equal to −Id4. Therefore, the only point with
nontrivial parabolic filtration is∞ and the only filtrands are of weight 1/2. Thus equation (2.1)
and Lemma 2.1 give
degE1,2 + degE2,1 = −1.
By Theorem 2.7, the map θ1 : E2,1 → E1,2 ⊗ ΩP1(∆) has cokernel of length 2. Therefore,
degE2,1 − degE1,2 = 3.
Hence deg E2,1 = 1. This gives the correct b value in Theorem 2.11, which in turn allows us
to complete the corresponding row of the table in [23, Section 4]; we obtain that the Hodge
numbers of g∗L are (0, 1, 2, 1, 0).
Similarly, if we take g to be the degree 10 map ramified completely at 0 and ∞, then
degE1,2 + degE2,1 = 0.
In this case θ1 has cokernel of length 4, so
degE2,1 − degE1,2 = 6,
and therefore degE2,1 = 3. Once again, this allows us to complete the corresponding row of the
table in [23, Section 4]; we obtain that the Hodge numbers of g∗L are (0, 1, 3, 1, 0).
Using this, we may complete the calculations left unfinished by [3] and [23]. Table 3 is
a reproduction of the table from [23, Section 4], with all missing pieces completed; our new
results are highlighted in gray. Our notation is as follows. # is the case number from the table in
[23, Section 4], and the numbers (α1, α2, α3, α4) are the numbers determining the corresponding
hypergeometric differential equations
L = δ4 − t(δ + α1)(δ + α2)(δ + α3)(δ + α4).
The number d indicates that we have pulled back by the covering [t : s] 7→ [td : sd], to obtain
a VHS over P1 with Hodge bundles E3,0 of degree a and E2,1 of degree b. We then apply the
results of Section 2.4 to compute the Hodge numbers of the corresponding parabolic cohomology
groups in the final column.
Remark 5.2. Recently there has been a great deal of interest in studying the monodromy
subgroups of Sp(4,R) in the 14 hypergeometric examples of Doran and Morgan [12]. These
subgroups may be either arithmetic or non-arithmetic (more commonly called thin). Singh and
Venkataramana [28, 29] have proved that the monodromy subgroup is arithmetic in seven of
the 14 cases, and Brav and Thomas [1] have proved that it is thin in the remaining seven. The
arithmetic/thin status of each of the examples is given in the “Monodromy” column of Table 3.
Computations carried out in [13] suggest that the arithmetic/thin dichotomy for monodromy
groups of hypergeometric local systems is closely related to the value of the Lyapunov exponents
18 C.F. Doran, A. Harder and A. Thompson
of the corresponding variations of Hodge structure. Lyapunov exponents are dynamical invari-
ants associated to a flat vector bundle over a hyperbolic Riemann surface, that are computed
using the monodromy of the flat bundle as well as the metric structure on the hyperbolic curve.
The main goal of [13] is to study these Lyapunov exponents; [13, Lemma 6.3] (restated above
as Theorem 2.7) is a step towards this. In [13] it is conjectured that for the 14 hypergeometric
variations of Hodge structure of Doran and Morgan [12], the associated Lyapunov exponents are
closely related to the parabolic degrees of the bundles E3,0 and E2,1, provided that these degrees
satisfy a certain relation (see [13, Conjecture 6.4] and [15] for the higher rank case).
It is not clear whether there is any relation between the computations presented here and
Lyapunov exponents; however, it would be interesting to see whether there is any relation
between Lyapunov exponents and the asymptotic growth of degpar E
3,0 and degpar E
2,1 as the
degree d of the covering [t : s] 7→ [td : sd] grows. Such growth has been investigated in the rank 2
case by Kappes [24].
Finally we note that, according to work of Doran and Malmendier [11], most of Doran’s
and Morgan’s [12] 14 hypergeometric local systems can be obtained from hypergeometric local
systems of rank 3. These rank 3 hypergeometric local systems underlie variations of Hodge
structure corresponding to families of K3 surfaces, for which the Lyapunov exponents have been
computed by Filip [14]. It seems plausible that the iterated construction of [11] along with the
computations of [14] could allow one to compute Lyapunov exponents for the fourteen rank 4
hypergeometric local systems of [12].
Remark 5.3. We pose the question as to whether the above Hodge structures are reducible or
not. It is possible that the h2,2 value in each of the sets of Hodge numbers in the above table
all correspond to rational Hodge classes, thus induce normal functions on the corresponding
families of Calabi–Yau threefolds. The maps along which we have pulled back are rigid in the
sense that any deformation of this map will change the Hodge numbers, so this indeed seems
possible.
5.2 Calabi–Yau fourfolds fibred by mirror quintics
Finally, we will apply the same approach to classifying Calabi–Yau fourfolds fibred by Calabi–
Yau threefolds as we did with Calabi–Yau threefolds fibred by K3 surfaces. We do not obtain
concrete classification results, since modularity results for families of Calabi–Yau varieties of
higher dimension are scarce, but we are at least able to place bounds on the space of possibilities.
We start with the mirror quintic family of K3 surfaces. This family can be represented as
a compactification of the fibres of the Laurent polynomial
(x+ y + z + w + 1)5
xyzw
.
The periods of this family satsify the hypergeometric differential equation L
δ4 − t
(
δ +
1
5
)(
δ +
2
5
)(
δ +
3
5
)(
δ +
4
5
)
,
which has local monodromy matrices
T0 =
1 1 0 0
0 1 1 0
0 0 1 1
0 0 0 1
, T1 =
1 0 0 0
0 1 0 0
0 0 1 1
0 0 0 1
, T∞ =
ζ5 0 0 0
0 ζ2
5 0 0
0 0 ζ3
5 0
0 0 0 ζ4
5
,
Hodge Numbers from Picard–Fuchs Equations 19
# (α1, α2, α3, α4) d a b Hodge numbers Monodromy
1 (1/5, 2/5, 3/5, 4/5) 1 0 0 (0, 0, 0, 0, 0) Thin
2 0 0 (0, 0, 1, 0, 0)
5 1 2 (0, 0, 0, 0, 0)
10 2 4 (1, 1, 1, 1, 1)
2 (1/10, 3/10, 7/10, 9/10) 1 0 0 (0, 0, 0, 0, 0) Arithmetic
2 0 0 (0, 0, 1, 0, 0)
5 0 1 (0, 1, 2, 1, 0)
10 1 3 (0, 1, 3, 1, 0)
3 (1/2, 1/2, 1/2, 1/2) 1 0 0 (0, 0, 0, 0, 0) Thin
2 1 1 (0, 0, 0, 0, 0)
2k k k (k − 1, 0, 0, 0, k − 1)
4 (1/3, 1/3, 2/3, 2/3) 1 0 0 (0, 0, 0, 0, 0) Arithmetic
2 0 0 (0, 0, 1, 0, 0)
3 1 1 (0, 0, 0, 0, 0)
6 2 2 (1, 0, 1, 0, 1)
5 (1/3, 1/2, 1/2, 2/3) 1 0 0 (0, 0, 0, 0, 0) Thin
6 1 2 (0, 0, 2, 0, 0)
6 (1/4, 1/2, 1/2, 3/4) 1 0 0 (0, 0, 0, 0, 0) Thin
4 1 2 (0, 0, 0, 0, 0)
8 2 4 (1, 1, 0, 1, 1)
7 (1/8, 3/8, 5/8, 7/8) 1 0 0 (0, 0, 0, 0, 0) Thin
2 0 0 (0, 0, 1, 0, 0)
4 0 1 (0, 1, 1, 1, 0)
8 1 3 (0, 1, 1, 1, 0)
8 (1/6, 1/3, 2/3, 5/6) 1 0 0 (0, 0, 0, 0, 0) Arithmetic
2 0 0 (0, 0, 1, 0, 0)
6 1 2 (0, 0, 1, 0, 0)
9 (1/12, 5/12, 7/12, 11/12) 1 0 0 (0, 0, 0, 0, 0) Thin
2 0 0 (0, 0, 1, 0, 0)
3 0 1 (0, 1, 0, 1, 0)
4 0 1 (0, 1, 1, 1, 0)
6 0 2 (0, 2, 1, 2, 0)
12 1 5 (0, 3, 1, 3, 0)
10 (1/4, 1/4, 3/4, 3/4) 1 0 0 (0, 0, 0, 0, 0) Arithmetic
2 0 0 (0, 0, 1, 0, 0)
4 1 1 (0, 0, 1, 0, 0)
8 2 2 (1, 0, 3, 0, 1)
11 (1/6, 1/4, 3/4, 5/6) 1 0 0 (0, 0, 0, 0, 0) Arithmetic
2 0 0 (0, 0, 1, 0, 0)
12 1 3 (0, 1, 5, 1, 0)
12 (1/4, 1/3, 2/3, 3/4) 1 0 0 (0, 0, 0, 0, 0) Arithmetic
2 0 0 (0, 0, 1, 0, 0)
3 0 1 (0, 1, 0, 1, 0)
12 1 4 (0, 2, 3, 2, 0)
13 (1/6, 1/6, 5/6, 5/6) 1 0 0 (0, 0, 0, 0, 0) Arithmetic
2 0 0 (0, 0, 1, 0, 0)
3 0 0 (0, 0, 2, 0, 0)
6 1 1 (0, 0, 3, 0, 0)
14 (1/6, 1/2, 1/2, 5/6) 1 0 0 (0, 0, 0, 0, 0) Thin
3 0 1 (0, 1, 0, 1, 0)
6 1 3 (0, 1, 0, 1, 0)
Table 3. Invariants for families of Calabi–Yau threefolds.
20 C.F. Doran, A. Harder and A. Thompson
and Riemann scheme
0 1 ∞
0 0 1/5
0 1 2/5
0 1 3/5
0 2 4/5
.
We can ask: along which maps g : P1 → P1 does the pull-back g∗L have Hodge number h0,4 = 1?
In other words, when can the fourfold total space obtained from the mirror quintic family be
a Calabi–Yau fourfold?
We have all the tools to perform this calculation. Let us take a map g : P1 → P1 of degree d.
Let k, `, and m denote the numbers of points over 0, ∞, and 1 respectively and let [y1, . . . , y`]
denote the partition of d encoding the ramification profile over∞. Finally, let `0 be the number
of values of i such that 5 - yi. We compute the Hodge numbers of the parabolic cohomology
associated to g∗L.
Proposition 5.4. The degrees of the Hodge bundles of the variation of Hodge structure on g∗L
are given by
degE3,0 =
1
2
(
d− `0 −
∑̀
i=1
(
b3yi
5 c − b
2yi
5 c
))
−
∑̀
i=1
(
b2yi
5 c − b
yi
5 c
)
degE2,1 =
1
2
(
d− `0 −
∑̀
i=1
(
b3yi
5 c − b
2yi
5 c
))
,
and the equations degE3,0 + degE0,3 = −`0 and degE2,1 + degE1,2 = −`0.
Remark 5.5. From this data, one can compute all Hodge numbers of the parabolic cohomology
groups, by applying the results of Section 2.4.
Proof. By Theorem 2.7, the map θ1 : E2,1 → E1,2 ⊗ ΩP1(∆) has cokernel of length
r +
∑̀
i=1
(
b3yi
5 c − b
2yi
5 c
)
− (`− `0)
(note that (`− `0) counts the number of values of i so that 5 | yi). Thus
degE2,1 − degE1,2 = k + `+m− 2− r −
∑̀
i=1
(
b3yi
5 c − b
2yi
5 c
)
.
Applying the Riemann–Hurwitz formula, we see that
2d− 2 =
∑
p
(ep − 1) = r + (d− k) + (d− `) + (d−m), (5.1)
which implies that d = k + `+m− r − 2. Therefore,
degE2,1 − degE1,2 = d−
∑̀
i=1
(
b3yi
5 c − b
2yi
5 c
)
.
Hodge Numbers from Picard–Fuchs Equations 21
Equation (2.1) and Lemma 2.1 also give that degE3−i,i + degEi,3−i = −`0. So we conclude
that
degE2,1 =
1
2
(
d− `0 −
∑̀
i=1
(
b3yi
5 c − b
2yi
5 c
))
.
We also have that the cokernel of the map θ0 : E3,0 → E2,1 ⊗ ΩP1(∆) has length
r + d+
∑̀
i=1
(
b2yi
5 c − b
yi
5 c
)
− (`− `0),
thus
degE3,0 − degE2,1 = k + `+m− 2− r − d−
∑̀
i=1
(
b2yi
5 c − b
yi
5 c
)
= −
∑̀
i=1
(
b2yi
5 c − b
yi
5 c
)
,
where the second line follows by applying equation (5.1). Adding the expression for degE2,1
obtained above, we get
degE3,0 =
1
2
(
d− `0 −
∑̀
i=1
(
b3yi
5 c − b
2yi
5 c
))
−
∑̀
i=1
(
b2yi
5 c − b
yi
5 c
)
,
which completes the proof of the theorem. �
Since we have that degE3,0 + degE0,3 = −`0, it follows that
degE0,3 = −1
2
(
d+ `0 −
∑̀
i=1
(
b3yi
5 c − b
2yi
5 c
))
+
∑̀
i=1
(
b2yi
5 c − b
yi
5 c
)
.
Therefore,
Corollary 5.6. The Hodge structure on g∗L is Calabi–Yau if and only if
1
2
(
d+ `0 −
∑̀
i=1
(
b3yi
5 c − b
2yi
5 c
))
−
∑̀
i=1
(
b2yi
5 c − b
yi
5 c
)
= 2.
Remark 5.7. Note that the value of h0,3 of the parabolic cohomology only depends on the
structure of ramification over the point ∞, just as in the case of quartic threefolds.
A consequence of this corollary is that there is a finite and enumerable set of values for
[y1, . . . , y`] so that g∗L is Calabi–Yau. This set strictly contains the possible ramification data
over∞ under which the total space of the pullback of the family of mirror quintics is Calabi–Yau.
The list of all values of [y1, . . . , y`] are as follows:
[1, 1], [1, 2], [1, 3], [1, 4], [1, 5], [2, 2], [2, 3], [2, 4], [2, 5],
[3, 3], [3, 4], [3, 5], [4, 4], [4, 5], [5, 5], [6], [7], [8], [9], [10].
We expect that a proper subset of these ramification indices should correspond to maps along
which we can pull back the family of mirror quintics and get a variety that admits a smooth (or
terminal) Calabi–Yau resolution.
One can also check that the only cases in which the Hodge number h3,0 of the parabolic
cohomology is 0 are the cases where ` = 1 and y1 ∈ {1, 2, 3, 4, 5}.
22 C.F. Doran, A. Harder and A. Thompson
Remark 5.8. Similar calculations can presumably be completed for any family of Calabi–
Yau threefolds realising one of the fourteen hypergeometric variations of Hodge structure of
type (1, 1, 1, 1) classified by Doran and Morgan [12]. This includes the so-called “mirror twin”
families [12, Section 3], which have the same Picard–Fuchs equations and R-VHS’s as some
well-known toric examples, but have different underlying Z-VHS’s and Hodge number h1,1. In
particular, the calculations above apply equally to the “quintic mirror twin” family, which shares
its Picard–Fuchs equation and R-VHS with the quintic mirror example above.
Acknowledgements
C.F. Doran (University of Alberta) was supported by the Natural Sciences and Engineering
Research Council of Canada, the Pacific Institute for the Mathematical Sciences, and the Visiting
Campobassi Professorship at the University of Maryland.
A. Harder (University of Miami) was partially supported by the Simons Collaboration Grant
in Homological Mirror Symmetry.
A. Thompson (University of Warwick/University of Cambridge) was supported by the Engi-
neering and Physical Sciences Research Council programme grant Classification, Computation,
and Construction: New Methods in Geometry.
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1 Introduction
1.1 Structure of this paper
2 Background and important results
2.1 Variations of Hodge structure
2.2 Picard–Fuchs equation of a variation of Hodge structure
2.3 Fuchsian ODE's
2.4 Important results
2.5 Base change and characteristic exponents
3 Families of elliptic curves
4 Families of K3 surfaces
4.1 Base change of a hypergeometric family
5 Families of Calabi–Yau threefolds
5.1 Inhomogeneous Picard–Fuchs equations and Calabi–Yau threefolds
5.2 Calabi–Yau fourfolds fibred by mirror quintics
References
|
| id | nasplib_isofts_kiev_ua-123456789-148559 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:24:16Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Doran, C.F. Harder, A. Thompson, A. 2019-02-18T15:44:19Z 2019-02-18T15:44:19Z 2017 Hodge Numbers from Picard-Fuchs Equations / C.F. Doran, A. Harder, A. Thompson // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 31 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14D07; 14D05; 14J32 DOI:10.3842/SIGMA.2017.045 https://nasplib.isofts.kiev.ua/handle/123456789/148559 Given a variation of Hodge structure over P¹ with Hodge numbers (1,1,…,1), we show how to compute the degrees of the Deligne extension of its Hodge bundles, following Eskin-Kontsevich-Möller-Zorich, by using the local exponents of the corresponding Picard-Fuchs equation. This allows us to compute the Hodge numbers of Zucker's Hodge structure on the corresponding parabolic cohomology groups. We also apply this to families of elliptic curves, K3 surfaces and Calabi-Yau threefolds. 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.
 C.F. Doran (University of Alberta) was supported by the Natural Sciences and Engineering
 Research Council of Canada, the Pacific Institute for the Mathematical Sciences, and the Visiting
 Campobassi Professorship at the University of Maryland.
 A. Harder (University of Miami) was partially supported by the Simons Collaboration Grant
 in Homological Mirror Symmetry.
 A. Thompson (University of Warwick/University of Cambridge) was supported by the Engineering and Physical Sciences Research Council programme grant Classification, Computation,
 and Construction: New Methods in Geometry. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Hodge Numbers from Picard-Fuchs Equations Article published earlier |
| spellingShingle | Hodge Numbers from Picard-Fuchs Equations Doran, C.F. Harder, A. Thompson, A. |
| title | Hodge Numbers from Picard-Fuchs Equations |
| title_full | Hodge Numbers from Picard-Fuchs Equations |
| title_fullStr | Hodge Numbers from Picard-Fuchs Equations |
| title_full_unstemmed | Hodge Numbers from Picard-Fuchs Equations |
| title_short | Hodge Numbers from Picard-Fuchs Equations |
| title_sort | hodge numbers from picard-fuchs equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148559 |
| work_keys_str_mv | AT dorancf hodgenumbersfrompicardfuchsequations AT hardera hodgenumbersfrompicardfuchsequations AT thompsona hodgenumbersfrompicardfuchsequations |