
THE FIELDS
INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES 
Thematic
Program on CalabiYau Varieties: Arithmetic, Geometry
and Physics
November
1822, 2013
Workshop on Hodge Theory in String Theory
a joint workshop with PIMS CRG Program “Geometry
and Physics”
Principal Organizers: Charles F. Doran, David Morrison,
Radu Laza, Johannes Walcher.




November 18

November 19

November 20

November 21

November 22

9:30–10.30






11:00–12:00






14:00–15:00






15:30–16:30






17:00–19:00

Reception





Speaker 
Title and Abstract 
Ballard, Matt 
Griffiths’ residue theorem via LandauGinzburg models
I will discuss how Griffiths’ residue theorem can be
understood from the sigmamodel/LGmodel correspondence. This is joint
work with David Favero (Alberta) and Ludmil Katzarkov (Miami/Vienna).

Charles, Francois 
Families of rational curves on holomorphic symplectic fourfolds
It has been shown by Bogomolov–Mumford and Mori–Mukai
that projective K3 surfaces contain ample rational curves. We construct
uniruled divisors and rational surfaces on every projective irreducible
holomorphic symplectic fourfold of $K3^[2]$ type. As a consequence,
we construct a canonical zerocycle $'a$ la Beauville–Voisin on
any such fourfold.
This is joint work with Gianluca Pacienza.

Dettweiler, Michael 
Construction of differential equations of Calabi–Yau type and
Hodge theory
In the work of Reiter and Bogner, many differential equations
of CalabiYau type are constructed using additive and multiplicative
convolution (= Hadamard product). We recall their work and describe
an algorithm, developed jointly by Claude Sabbah and the speaker, on
how to determine the basic underlying local and global Hodge numerical
data for the variations of Hodge structures underlying these differential
equations of CalabiYau type.

Doran, Chuck 
Landau–Ginzburg models of Fano threefolds and moduli spaces
of K3 surfaces
To a Fano threefold equipped with a complexified Kaehler class,
we may associate a LandauGinzburg potential whose fibers over the complex
line are K3 surfaces. In this talk, wewill illustrate, via examples,
howthe birational geometry of Fano threefolds is captured by the geometry
of special families of curves in themoduli spaces of lattice polarized
K3 surfaces whichareDolgachevNikulin mirror to the anti canonical
K3 surfaces in the Fano threefold. We will also discuss anapplication
to the geometric construction ofcertainCalabiYau threefolds whose moduli
space isthe thricepunctured sphere.

Garbagnati, Alice 
Calabi–Yau threefolds of Borcea–Voisin type
The BorceaVoisin construction is a way to produce Calabi–Yau
3folds as crepant resolutions of quotients (S×E)/(Z/2Z) where
S is a K3 surface, E is an elliptic curve and Z/2Z acts diagonally on
S×E. Several generalizations of this construction were considered
in the last years. Here we consider Calabi–Yau 3folds which are
crepant resolutions of quotients (S × E)/Z/nZ where n = 2, 3,
4, 6 and, as before, S is a K3, E is an elliptic curve and Z/nZ acts
diagonally on S ×E. This imposes restrictions both on the elliptic
curve E and on the K3 surface S. We study the K3 surfaces involved in
this construction, we describe explicitly certain crepant resolutions
of (S ×E)/(Z/nZ), we compute the Hodge numbers of the Calabi–Yau
obtained. Some of the Calabi–Yau constructed are ”new”
and some of them lie in families without maximal unipotent monodromy.
In certain cases one proves that the variation of the Hodge structures
of the families of Calabi–Yau considered is essentially the variation
of the Hodge structures of families of curves. Moreover, by construction,
the Calabi–Yau 3folds obtained admit an (almost) elliptic fibration
which is isotrivial. We describe this fibration and give a Weierstrass
equation in certain cases. Some of the results presented are obtained
in collaboration with Bert van Geemen, otherswith Andrea Cattaneo.

Katzarkov, Ludmil 
Mixed Hodge structures and phantoms
On this talk we will look at some classical examples from
the point of view of category theory and Homological Mirror Symmetry.

Kerr, Matt 
Algebraic cycles and local quantum cohomology
In this talk, based on joint work with C. Doran, I will describe
some possible Amodel interpretations of variations of mixed Hodge structure
arising in local and open mirror symmetry. The discussion will focus
on examples arising in the work of Hosono and Morrison/Walcher, and
will highlight in each case interesting questions about homological
mirrors of cycleclass maps.

Malmendier, Andreas
Lecture Notes 
Multiparameter families of K3 surfaces from SeibergWitten curves
and hypergeometric functions.
In my talk I will generalize Sen's procedure by constructing all 2parameter
families of latticepolarized K3 surfaces that can be obtained from
extremal rational elliptic surfaces through a quadratic twist. I will
show that for all of these families the PicardFuchs system governing
the K3periods are obtained by an integral transform of a differential
equation of hypergeometric or Heun type, and that in fact the K3periods
have an interpretation as modular forms and solutions to a GKZ system.
If time permits I will also explain how further generalization of this
procedure naturally leads to K3 surfaces admitting double covers onto
P2 branched along a plane sextic curve. (This is joint work with Chuck
Doran, University of Alberta)

Morrison, David 
Variations of Hodge structure, GromovWitten invariants, and the
Gamma class
The original mirror symmetry predictions of GromovWitten
invariants of CalabiYau threefolds relied heavily on the behavior
of a degenerating variation of Hodge structure near the boundary of
Calabi–Yau moduli space. This led to a definition in the early
1990’s of the ”Avariation of Hodge structure”: a degenerating
variation of Hodge structure directly constructed from the GromovWitten
invariants themselves.
Recently, there have been advances in the physical study of the ”Amodel”
(the physical theory leading to GromovWitten invariants), which have
revealed that one aspect of the original definition of AVHS needs clarification
and modification. The modification involves the Gamma class, a characteristic
class closely related to the Gamma function.
We will explain this modification, and discuss some interesting examples.
Clemens–Schmid and mixed Hodge structures in string theory
The theory of mixed Hodge structures and the Clemens–Schmid
exact sequence have seen
several recent applications in string theory, which I shall survey.

Movasati, Hossein 
A common frame work for automorphic forms and topological partition
functions
Classical modular forms and in general automorphic forms enjoy
qexpansions with fruitful applications in different branches of mathematics.
From another side we have qexpansions coming from the Bmodel computations
of mirror symmetry which, in general, are believed to be new functions.
In this talk I will present a common algebrogeometric framework for
all these qexpansions. This is based on the moduli of varieties with
a fixed topological data and enhanced with a basis of the algebraic
de Rham cohomology, compatible with the Hodge filtration and with a
constant intersection matrix. In our way, we will also enlarge the algebra
of automorphic forms to a bigger algebra which is closed under canonical
derivations. I will mainly discuss three examples: 1. Elliptic curves
and classical modular forms, 2. Principally polarized abelian varieties,
lattice polarized K3 surfaces and Siegel modular forms 3. Mirror quintic
CalabiYau varieties, Yukawa coupling and topological partition functions.

Mustata, Anca 
The Dwork pencil of quintic threefolds
In this talk we will review existing results and open questions
related to Dwork pencil of quintic threefolds and its quotient, the
mirror quintic. This family of quintics has proven a particularly fertile
testing ground from the beginnings of mirror symmetry.

Pearlstein, Greg 
Naive limits of Hodge structure
Traditionally, one computes the asymptotic periods of a variation
of Hodge structure with respect to the canonical extension. For Hodge
structures of high level, there is also interesting information in the
naive limit filtration. This talk is based on joint work with M. Kerr.

Peters, Chris 
Two isomorphic classical domains and related geometric moduli spaces
The classical isomorphism of simple Lie groups
U(2, 2)/center $\simeq$ O(2, 4)/center
leads to a biholomorphic isomorphism between the corresponding homogeneous
domains
$H_2$ = U(2, 2)/U(2) × U(2) $\simeq$ D4 = O(2, 4)/O(2) ×
O(4).
The domains have complex dimension 4. The first, $H_2$ parametrizes
principally polarized Abelian 4folds with an extra involution of order
4 and the second, $D_4$ parametrizes K3surfaces that are double covers
of the plane branched in 6 lines in general position. The isomorphism
can be explained using Hodge theory. To obtain the true moduli spaces
one divides out by the appropriate discrete groups. Certain divisors
of these moduli spaces studied previously on the Abelian 4fold side
have explicit descriptions on the K3side. Finally, there is an intriguing
relation with the KugaSatake correspondence associated to ”generic”
points in these moduli spaces as well as their divisors.
This is a report of recent joint work with Giuseppe Lombardo and Matthias
Schütt.

Sacca, Giulia 
CalabiYau manifolds and relative Jacobians of linear systems on
surfaces with trivial Kodaira dimension
Let X be a surface whose canonical bundle is (non trivial)
torsion of degree 2, and let C be a linear system on X. I will show
how the relative compactified Jacobian of C is an (odd dimensional)
CalabiYau manifold. This result is conditional to a very natural assumption
that holds for low values of the genus of the linear system, and that
is expected to hold in general. After discussing this assumption, I
will focus about some aspects of the topology of these relative compactified
Jacobians.

Usui, Sampei
Lecture Notes

Studies of closed/open mirror symmetry for quintic threefolds through
log mixed Hodge theory
We try to understand closed and open mirror symmetry for quintic
threefolds in the framework of the fundamental diagram, obtained by
the joint work of Kato, Nakayama, and Usui, which relates various compactifications
of classifying space of mixed Hodge structures.

Voisin, Claire 
Varieties of power sums and divisors on the moduli space of cubic
fourfolds
This is a joint work with K. Ranestad. To a cubic fourfold
X is associated its variety of lines, which is known by Beauville and
Donagi to be a hyperKhler fourfold whose Hodge structure on degree
2 cohomology is isomorphic to the Hodge structure on degree 4 cohomology
of X. Iliev and Ranestad associated to X another hyperKhler fourfold,
constructed as the variety of powersums of X. We show that for this
second hyperKhler fourfold, there is (for general X) no nontrivial
morphism of Hodge structures from the Hodge structure on its degree
2 cohomology to the Hodge structure on degree 4 cohomology of X.

Walcher, Johannes 
On 2functions and their framing
2functions are defined by an integrability condition with
respect to the action of the Frobenius endomorphism on formal power
series with algebraic coefficients. They play a role in (open string)
mirror symmetry, and perhaps in other contexts as well. Among their
nontrivial elementary properties is the stability under the framing
transformation, which can be proven in several different ways. Based
on joint work with A. Schwarz and V. Vologodsky.

Zuo, Kang 
On Shimura curves in Torelli locus of hyperelliptic curves
In my talk I shall report my recent joint work with Xin LU
We show that there do not exist Shimura curves contained in generically
in the Torelli locus of hyperelliptic curves of genus g > 7. We also
present examples of such Shimura curves for g = 3 or 4.

Participant List
Full Name 
University/Affiliation 
Ballard, Matthew 
University of South Carolina 
Ceballos, Cesar 
York University 
Charles, Francois 
Massachusetts Institute of Technology 
Dettweiler, Michael 
University Bayreuth 
Doran, Charles 
University of Alberta 
Faber, Eleonore 
University of Toronto 
Filippini, Sara Angela 
Fields Institute 
Findleton, Suzanne 
Queen's University 
Garbagnati, Alice 
Universita degli Studi di Milano 
GarciaRaboso, Alberto 
University of Toronto 
Halic, Mihai 

Harder, Andrew 
University of Aberta 
Katzarkov, Ludmil 
University of California, Irvine 
Keast, Ryan 
Washington University in St. Louis 
Kelly, Tyler 
University of Pennsylvania 
Kerr, Matt 
Washington University in St. Louis 
Laza, Radu 
Stony Brook University 
Luk, Kevin 
University of Toronto 
Malmendier, Andreas 
Colby College 
Molnar, Alexander 
Queen's University 
Morrison, David R. 
U. C. Santa Barbara 
Movasati, Hossein 
Instituto Nacional de Matemática Pura e Aplicada 
Mustata, Anca 
University College Cork 
Overholser, Douglas 
Fields Institute 
Pearlstein, Gregory 
Texas A&M 
Perunicic, Andrija 
Fields Institute 
Peters, Chris 
Universite Grenoble I 
Rayan, Steven 
University of Toronto 
Rose, Simon 
Fields Institute 
Ruddat, Helge 
Universität Mainz 
Sacca, Giulia 
Stony Brook University 
Soloviev, Fedor 
University of Toronto 
Thompson, Alan 
Fields Institute 
Usui, Sampei 
Osaka University 
van Garrel, Michel 
Fields Institute 
Voisin, Claire 
CNRS 
Walcher, Johannes 
McGill University 
Xu, Jinxing 
University of Science and Technology of China 
Yui, Noriko 
Queen's University 
Zhang, Mingwei 
University of Science and Technology of China 
Zhang, Zheng 
Stony Brook University 
Zhu, Yuecheng 
University of Texas at Austin 
Zuo, Kang 
University Mainz 
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