Master class: Around Torelli's theorem for K3 surfaces

Arithmetic, Geometric and Dynamical aspects

October 28-November 1 2013, Institut de Recherche de Mathématique Avancée, Strasbourg

New: some lecture notes (in french) for Samuel Boissière's lectures are available: here.


Compact complex surfaces

Hodge Theory

Deformations of complex manifolds

Main lectures

Torelli theorem for K3 surfaces (Samuel Boissière)

The Torelli theorem for K3 surfaces relates the biholomorphic transformations of K3 surfaces to an algebraic invariant associated to this family of complex surfaces: their second cohomology group that, besides its weight two polarized Hodge structure, admits a structure of even unimodular lattice. A classical application of this theorem is the construction of automorphisms of K3 surfaces from isometries of this lattice satisfying some natural conditions. In these lectures, I'll explain this theorem by using some classical families of K3 surfaces as quartic surfaces and Kummer surfaces. I'll give some fundamental applications of this theorem related to the structure of moduli spaces of K3 surfaces and their automorphisms.

Hyperkähler manifolds and counter-examples to Torelli's theorem (Christian Lehn, Gianluca Pacienza and Pierre Py)

We will give a gentle introduction to hyperkähler manifolds, which are a natural generalization of K3 surfaces to higher dimensions. We will first explain where they sit in the classification of kähler manifolds, namely, we will describe Beauville's decomposition theorem for kähler manifolds with vanishing first Chern class (according to which hyperkähler manifolds appear as elementary factors). Second, we will discuss in details the counter-examples to the global Torelli theorem given by Debarre and Namikawa. This will make a link with the title of the master class and will prepare the students to Verbitsky's research talks.

The crystalline Torelli theorem for supersingular K3 surfaces, and unirationality of K3 surfaces (Christian Liedtke)

Over algebraically closed fields of characteristic different from $2,3$, surfaces with trivial canonical sheaves are K3 surfaces or Abelian surfaces. In this course, we will first introduce K3 surfaces in arbitrary characteristic, and determine their algebraic deRham cohomology. Next, we will see that crystalline cohomology (no prior knowledge assumed) is the ``right'' replacement for singular cohomology in positive characteristic. Then, we will look at one particular class of K3 surfaces more closely, namely, supersingular K3 surfaces. By definition, these have Picard rank $22$ (note: in characteristic zero, at most rank $20$ is possible) and form $9$-dimensional moduli spaces. For supersingular K3 surfaces, we will see that there exists a period map and a Torelli theorem in terms of crystalline cohomology. As an application of the crystalline Torelli theorem, we will show that a K3 surface is supersingular if and only if it is unirational.

Dynamical properties of K3 surfaces automorphisms (Julien Grivaux)

We will start by giving an overview of dynamical properties of automorphisms of complex compact surfaces. Among all automorphisms, those with positive topological entropy are rare but are also the most interesting, both from the point of view of complex dynamics or algebraic geometry. We will focus on the case of K3 surfaces, and prove that it is possible to construct automorphisms of positive entropy that are ergodic for the Lebesgue measure on Kummer surfaces. On the contrary, as a main application of Torelli's theorem, we will prove McMullen's result about the existence of automorphisms of positive entropy admitting a 2-dimensional Siegel disk on non-projective K3's, which is an obstruction to ergodicity.

Research talks by Misha Verbitsky

Talk 1: Global Torelli theorem for hyperkähler manifolds

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hyperkähler manifold M, showing that it is commensurable to an arithmetic subgroup in SO(3, b2-3). A Teichmuller space of M is a space of complex structures on M up to isotopies. We define a birational Teichmuller space by identifying certain points corresponding to bimeromorphically equivalent manifolds, and show that the period map gives an isomorphism of the birational Teichmuller space and the corresponding period space SO(b2-3, 3)/SO(2) x SO(b2 -3, 1). We use this result to obtain a Torelli theorem identifying any connected component of birational moduli space with a quotient of a period space by an arithmetic subgroup. When M is a Hilbert scheme of n points on a K3 surface, with n-1 a prime power, our Torelli theorem implies the usual Hodge-theoretic birational Torelli theorem (for other examples of hyperkähler manifolds the Hodge-theoretic Torelli theorem is known to be false).

Talk 2: Ergodic complex structures

Let M be a compact complex manifold. The corresponding Teichmuller space Teich is a space of all complex structures on M up to the action of the group of isotopies. The group C of connected components of the diffeomorphism group (known as the mapping class group) acts on Teich in a natural way. An ergodic complex structure is the one with a C-orbit dense in Teich. Let M be a complex torus of complex dimension greater or equal to 2 or a hyperkahler manifold with second Betti number b2 greater than 3. We prove that M is ergodic, unless M has maximal Picard rank (there is a countable number of such M). This is used to show that all hyperkähler manifolds are Kobayashi non-hyperbolic.