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School of algebraic geometry

Program

24-28 septembre 2012, station biologique de Roscoff

We arranged for three minicourses:

- Julien Grivaux (Marseille)

Introduction to variations of Hodge structures

- Samuel Grushevsky (Stony Brook)

Curves, divisors, and birational geometry of the moduli space of curves

- Dimitri Zvonkine (Jussieu)

An introduction to the intersection theory of the moduli spaces of stable curves

Obviously, the goal is to have minicourses oriented to all but
algebraic geometers: the audience corresponds to

specialists in
dynamics, and not in algebraic geometry.

Below is a more detailed program.

MONDAY

10:00 - 10:30 coffee break/accueil des participants

10:30 - 12:00 Samuel Grushevsky 1/4

13h00 lunch

14:30 - 15:30 Julien Grivaux 1/5

15:30 - 15:45 coffee break

15:45 - 16:45 Dimitri Zvonkine 1/5

THUESDAY

10:00 - 11:00 Julien Grivaux 2/5

11:00 - 11:30 coffee break

11:30 - 12:30 Dimitri Zvonkine 2/5

13h00 lunch

14:30 - 16:00 Samuel Grushevsky 2/4

WEDNESDAY

10:00 - 11:00 Julien Grivaux 3/5

11:00 - 11:30 coffee break

11:30 - 12:30 Dimitri Zvonkine 3/5

13h00 lunch

14h30 FREE (walk on l'Ile de Batz)

THURSDAY

10:00 - 11:00 Julien Grivaux 4/5

11:00 - 11:30 coffee break

11:30 - 12:30 Dimitri Zvonkine 4/5

13h00 lunch

14:30 - 15:30 Samuel Grushevsky 3/4

19h dinner

FRIDAY

10:00 - 11:00 Julien Grivaux 5/5

11:00 - 11:30 coffee break

11:30 - 12:30 Samuel Grushevsky 4/4

13h00 lunch

14:30 - 15:30 Dimitri Zvonkine 5/5

15:30 coffee break

** Julien Grivaux (Marseille)
Introduction to variations of Hodge structures **

A Hodge structure is a geometric structure underlying the cohomology of any smooth complex projective variety. It is a fascinating object because it intertwins deeply arithmetic and geometry. For instance, the simplest Hodge structures (those being of weight one) correspond to complex tori. In this course, our aim is to give an introduction to the theory of variations of Hodge structures, which are holomorphic families of Hodge structures parameterized by a complex manifold. As an application of this theory, we will be able to prove the following theorem (which is a particular case of a more general result of Eskin-Kontsevich-Zorich): if C is a Teichmüller curve, the sum of the positive Lyapounov exponents of the Teichmüller flow on C is a rational number.

Since the public is not assumed to be acquainted with algebraic geometry, we will introduce all the necessary objects in the first lectures before dealing with variations of Hodge structure. The plan of the lectures will be the following:

Lecture 1: Introduction to sheaves.

Definitions, operations and cohomology. Examples.

Lecture 2: The Riemann-Hilbert correspondence on curves.

Local systems, flat connexions and monodromy. Regular connexions and residues.

Deligne's extension theorem.

Lecture 3: Hodge structures associated with curves.

Complex tori and abelian varieties. Polarized Hodge structures of weight one.

Jacobians.

Lecture 4: Variations of Hodge structures (I).

Period domain and period mapping. The monodromy theorem.

Lecture 5: Variations of Hodge structures (II).

The nilpotent orbit theorem. Extension of the Hodge bundle.

Deligne's semisimplicity theorem.

** Samuel Grushevsky (Stony Brook)
Curves, divisors, and birational geometry of the moduli space of curves.**

In algebraic geometry, birational invariants of an algebraic variety are essentially those properties that are preserved by isomorphisms defined outside a subvariety. Perhaps the most important such invariant is the Kodaira dimension, which measures the growth of the dimension of space of pluricanonical forms (that is, the space of sections of tensor powers of the line bundle of top degree holomorphic forms). In particular the Kodaira dimension detects whether a variety can be parametrized by the projective space: whether there exists a global coordinate system defined outside of a subvariety. In these lectures we will discuss the birational geometry of the moduli space of curves.

It turns out that birational geometry is largely concerned with classes of divisors (complex codimension one subvarieties) and curves on a variety, and the hands-on goal of this class is to explain and demonstrate many of the techniques used for computing divisor classes on the moduli space of curves, in particular using modular forms, test curves, and Grothendieck-Riemann-Roch. Many examples of divisor classes, especially in low genus, will be computed. We will also discuss the notion of slope of a divisor, its significance, and the known results, open problems, and ongoing bets regarding it. Divisor classes on the moduli space of abelian differentials will be mentioned. The Deligne-Mumford compactification will be discussed intuitively, but not constructed rigorously, and stack considerations will be avoided.

**Dimitri Zvonkine (Jussieu)
An introduction to the intersection theory of the moduli spaces of stable curves.**

The intersection theory of an algebraic variety M looks for answers to the following questions: What are the interesting cycles (algebraic subvarieties) of M and what cohomology classes do they represent? What are the interesting vector bundles over M and what are their characteristic classes? Can we describe the full cohomology ring of M and identify the above classes in this ring? Can we compute their intersection numbers?

In the case of moduli space, the full cohomology ring is still unknown. We are going to study its subring called the "tautological ring'' that contains the classes of most interesting cycles and the characteristic classes of most interesting vector bundles.

To give us a sense of purpose, we assume the following goal: after following the lectures, one should be able to write a computer program evaluating all intersection numbers between the tautological classes on the moduli space of stable curves. And to understand the foundation of every step of these computations. A program like that was first written by C. Faber, but our approach is a little different.

Here is a tentative plan of the lectures.

1. An informal introduction to moduli spaces of smooth and stable curves with some definitions and theorems and lots of examples, but no proofs.

2. Definition of the tautological cohomology classes on the moduli spaces. Simplest computations of intersection numbers.

3. Applying the Grothendieck-Riemann-Roch formula.

4. Reducing all intersection numbers to intersection numbers of psi-classes only. In this part we follow M. Kazarian's unpublished notes.

5. The ELSV formula: why certain intersection numbers on moduli spaces count factorizations of a given permutation into transpositions.