Winter School 2005 - Abstracts

  Voisin

We give an unified approach to the results of Gromov-Schoen and Delzant-Gromov and generalizes them. As an application we show that none of Thompson's group are Kahler.

In analogy with the holonomy groups in differential geometry, we define the notion of holonomy groups for a stable vector bundle. Then we illustrate the principle that either the holonomy is large or there is a clear geometric reason why it should be small. Joint work with Balaji.

The goal of this talk will be to explain the statement of the homological mirror symmetry conjecture in the case of Fano varieties, and how it can be verified on concrete examples. In this setting, the homological mirror symmetry conjecture predicts a correspondence between the derived category of coherent sheaves on a complex manifold and the derived category of Lagrangian vanishing cycles on its mirror. Here we will consider the special case of blowups of the projective plane, where the mirror correspondence can be determined in a very explicit manner.

An attempt will be made to sketch the main features of the proof of the Green-Griffiths conjecture for surfaces modulo resolution of singularities of ODEs on surfaces.

A multiplier is a function such that local a priori estimates for partial differential equations hold only after the test function is multiplied by it. The ideal sheaf consisting of multipliers identifies the location and the jet orders where local a priori estimates fail to hold. Solvability of a partial differential equation is reduced to algebraic conditions which force the multiplier ideal sheaf to be the structure sheaf. On the side of analysis, the method of multiplier ideal sheaves has been applied to problems such as the global regularity problem of the complex Neumann equation on pseudoconvex domains and the existence of Kaehler-Einstein metrics of Fano manifolds. On the side of algebraic geometry, the method of multiplier ideal sheaves has been successfully applied to solving, or making substantial progress towards solving, a number of long outstanding problems in algebraic geometry such as the Fujita conjecture, the effective Matsusaka big theorem, the deformational invariance of plurigenera, and the finite generation of canonical rings.

Stein manifolds have a canonical built-in symplectic geometry, called Weinstein structure, which is responsible, for instance, for the classification of Stein manifolds up to deformation. In the lectures I will review the results concerning this link and its applications. I will also discuss symplectic topology of (Wein-)Stein manifolds.

We will describe latest results on fundamental groups related to surfaces including outline of techniques, computational problems of the related BMT invariant and applications to cryptography.

I will present a certain alternative to the Hilbert scheme. This is a joint work with Allen Knutson.

Let B be a smooth complex curve and X a variety smooth and proper over C(B). Graber-Harris-Starr have shown that if X is geometrically rationally connected then X(C(B)) is nonempty. Building on work of Kollár-Miyaoka-Mori and others, we show that X satisfies weak approximation, at places of good reduction. We shall also discuss results at places of bad reduction in special cases like cubic surfaces. (joint work with Yuri Tschinkel)

I will explain a proof of the following statement: any Hilbert modular variety of dimension g>4 can't be included in the modular space of curves of genus g. Combined with my work on equidistribution on CM-points, this result implies the finiteness of certain curves with CM Jacobians. This is a report of a joint work in progress with Johan de Jong.

An automorphism of a K3 surface is called wild if its order is divisible by the characteristic of the ground field. I will explain why wild automorphisms occur only if the characteristic is less or equal than 11. I will discuss some examples of K3 surfaces which admit wild automorphisms and possible classification of finite groups of symplectic automorphisms containing wild automorphisms.

If X is a projective smooth geometrically irreducible variety defined over F _q , which is rationally connected over any algebraically closed field containing F _q , then its number of rational points is congruent to 1 modulo q (and in particular Fano varieties over a finite field carry a rational point as predicted by the Lang-Manin conjecture). If X is no longer smooth, then Kollár gave a beautiful example of a rationally connected variety over F _q without any point at all. In this talk, we discuss various conditions which force congruences for the number of rational points on projective singular varieties defined over finite fields. One theorem is that two theta divisors on an abelian variety defined over F _q carry the same number of points modulo q (joint with Berthelot-Bloch), as predicted by a consequence of a conjecture by Serre. Another is the number of rational points of a projective variety with rational singularities (to be appropriately defined), which is rationally connected over any algebraically closed field, is 1 modulo q.

Using mapping tori for surface diffeomorphisms in the Torelli group, a family of examples may be found of nonformal symplectic 4-folds satisfying the Hard Lefschetz theorem, with Massey products vanishing up to any wished for order. This is joint work with D. Kotschick.

It has been known for a long time that Hodge and de Rham cohomology of algebraic manifolds can be defined for non-commutative manifolds, too, as cyclic and Hochschild homology. Recently M. Kontsevich stated a rather precise conjecture as to when one should expect the degeneration of the non-commutative Hodge-to-de Rham spectral sequence. We will show how to prove this conjecture in some cases by the method of Deligne and Illusie. To do this, we will construct a non-commutative version of the Frobenius map and Cartier isomorphism.