SCIENTIFIC PROGRAMS AND ACTIVITIES

Griffiths’ residue theorem via Landau-Ginzburg models

I will discuss how Griffiths’ residue theorem can be understood from the sigma-model/LG-model correspondence. This is joint work with David Favero (Alberta) and Ludmil Katzarkov (Miami/Vienna).

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 zero-cycle $'a$ la Beauville–Voisin on any such fourfold.
This is joint work with Gianluca Pacienza.

Construction of differential equations of Calabi–Yau type and Hodge theory

In the work of Reiter and Bogner, many differential equations of Calabi-Yau 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 Calabi-Yau type.

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 Landau-Ginzburg 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 whichareDolgachev-Nikulin mirror to the anti- canonical K3 surfaces in the Fano threefold. We will also discuss anapplication to the geometric construction ofcertainCalabi-Yau threefolds whose moduli space isthe thrice-punctured sphere.

Calabi–Yau threefolds of Borcea–Voisin type

The Borcea-Voisin construction is a way to produce Calabi–Yau 3-folds 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 3-folds 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 3-folds 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.

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.

Algebraic cycles and local quantum cohomology

In this talk, based on joint work with C. Doran, I will describe some possible A-model 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 cycle-class maps.

Multi-parameter families of K3 surfaces from Seiberg-Witten curves and hypergeometric functions.

In my talk I will generalize Sen's procedure by constructing all 2-parameter families of lattice-polarized 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 Picard-Fuchs system governing the K3-periods are obtained by an integral transform of a differential equation of hypergeometric or Heun type, and that in fact the K3-periods 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)

Variations of Hodge structure, Gromov-Witten invariants, and the Gamma class

The original mirror symmetry predictions of Gromov-Witten invariants of Calabi-Yau three-folds 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 ”A-variation of Hodge structure”: a degenerating variation of Hodge structure directly constructed from the Gromov-Witten invariants themselves.
Recently, there have been advances in the physical study of the ”A-model” (the physical theory leading to Gromov-Witten invariants), which have revealed that one aspect of the original definition of A-VHS 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.

The theory of mixed Hodge structures and the Clemens–Schmid exact sequence have seen
several recent applications in string theory, which I shall survey.

A common frame work for automorphic forms and topological partition functions

Classical modular forms and in general automorphic forms enjoy q-expansions with fruitful applications in different branches of mathematics. From another side we have q-expansions coming from the B-model computations of mirror symmetry which, in general, are believed to be new functions. In this talk I will present a common algebro-geometric framework for all these q-expansions. 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 Calabi-Yau varieties, Yukawa coupling and topological partition functions.

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.

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.

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 4-folds with an extra involution of order 4 and the second, $D_4$ parametrizes K3-surfaces 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 4-fold side have explicit descriptions on the K3-side. Finally, there is an intriguing relation with the Kuga-Satake 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.

Calabi-Yau 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) Calabi-Yau 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 frame-work 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.

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 hyper-Khler 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 hyper-Khler fourfold, constructed as the variety of powersums of X. We show that for this second hyper-Khler fourfold, there is (for general X) no non-trivial morphism of Hodge structures from the Hodge structure on its degree 2 cohomology to the Hodge structure on degree 4 cohomology of X.

On 2-functions and their framing

2-functions 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 non-trivial 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.

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.