Abstracts of Talks
Kai Behrend (University of British Columbia)
When counting curves on a Calabi-Yau threefold, the first question to be answered is what one means by "curve". One possibility is "rank one torsion-free sheaf with trivial determinant". This count has been shown by the speaker to be a weighted Euler characteristic. We will explain a few attempts at constructing cohomologoy theories which give these weighted Euler characteristics as alternating sums of dimensions. paper one, paper two
Jim Bryan (University of British Columbia)
Get G be a finite subgroup of SO(3) and let Y be the resolution of the singularity C^3/G given by the G-Hilbert scheme. The classical McKay correspondence describes the classical geometry of Y in terms of the representation theory of G. We formulate and prove a quantum McKay correspondence: it describes the quantum geometry of Y, namely its Gromov-Witten theory, in terms of an ADE root system canonically associated to G.
Brendan Hassett (joint with Yuri Tschinkel)
Let F be a complex projective fourfold, deformation equivalent to the Hilbert scheme of a K3 surface. We have conjectured an explicit description for the ample cone of F in terms of the Hodge structure and the Beauville-Bogomolov form on the second cohomology group. This generalizes results for the ample cone of K3 surfaces arising out of the Torelli Theorem, but the existence of Mukai flops introduces new phenomena. We illustrate these with an example where the moving cone of F decomposes into a countably infinite number of chambers, each corresponding to a sequence of flops on F.
Mirror symmetry on Calabi-Yau manifolds implies that the generating functions of the Gromov-Witten invariants can be calculated in the B-model from the variation of the complex structure of a mirror manifold as encoded in the properties of the period integrals. We review the structure of the topological B-model and explain under which circumstances it can be completely solved. This lecture is supposed to be an introduction into B-model techniques. Connections of the latter to matrix models are outlined.
We focus first on the Calabi-Yau geometries in which the generating forms of the GW invariants are related to classical automorphic forms. Notably on K3 surfaces, del Pezzo surfaces embedded into Calabi-Yau threefolds and K3 fibered Calabi-Yau threefolds, in particular the Enriques Calabi-Yau. We explain how the structure of almost holomorphic automorphic forms arise from the holomorphic anomaly equation of the B-model. We then explore the polynomial structure of the B-model amplitudes in terms of rings of automorphic forms of jet unexplored modular groups that arise in the generic case.
In this talk, we will discuss various approaches to the Yau-Zaslow conjecture, especially for non-primitive cases.
The question we address is when a Heegner divisor in a locally sym metric variety of orthogonal or unitary type (for instance a Noether-Lefschetz locus in the moduli space of K3 surfaces of fixed degree) is the zero set of an automorphic form and therefore given by an infinite product expansion. Relevant is here a conjecture of Gritsenko and Nikulin.
I will consider pairs of topologically complementary moduli spaces of sheaves on a K3 surface. I will discuss conjectures and results concerning natural maps between spaces of sections of determinant line bundles on them.
We discuss the reduced GW-theory of K3 surfaces, both for rational and higher genus curves.
We describe a physical approach to deriving wall-crossing formulae for indices of BPS states in d=4, N=2 suprsymmetric theories based on multi-center boundstates. We describe applications of these formulae to generalized Donaldson-Thomas invariants and the OSV conjecture.
I will discuss open questions about moduli problems related to K3 surfaces: moduli of curves, sheaves, and of the K3s themselves. The lecture will attempt also to be an introduction to the workshop.
We explain that motivated by heterotic-type II duality, certain Gopakumar-Vafa invariants (and hence, conjecturally, Gromow-Witten invariants) for Calabi-Yau manifolds that admit a K3 fibration can be collected in a generating function. This function is in general an automorphic form determined by the topology of the fibration. In the class of K3 fibrations in toric varieties in which the Picard lattice of the fiber has rank one, we show how this automorphic form can be determined explicitly.
Yuri Tschinkel (joint with F. Bogomolov and M. Korotiaev)
will explain some constructions with hyperbolic curves and their Jacobians over finite fields.
Wei Zhang (Columbia University)
We will discuss a class of special algebraic cycles on Shimura varieties of orthogonal type and generating functions of their cycles classes in the Chow group. Two examples are 1) Heegner points on a modular curve and 2) Noether-Lefschetz loci on the moduli space of polarisable K-3 surfaces. Borcherds proved the modularity of generating functions of Heegner divisors based on the "singular Howe correspondence" of Harvey and Moore. Using an induction argument, I proved that generating functions of special cycle classes of any codimension, if convergent, must be Siegel modular forms. In some sense one may view the modularity as kind of duality between orthogonal groups and symplectic groups.