COURSE 1: Hyperbolicity properties of algebraic varieties

Jean-Pierre DEMAILLY (Université Grenoble Alpes, France)

The study of entire holomorphic curves drawn in projective algebraic varieties is intimately related to fascinating questions of value distribution theory and arithmetic geometry. One of the important unsolved questions is the Green-Griffiths-Lang conjecture which stipulates that for every projective variety $X$ of general type over $\mathbb{C}$ there exists a proper algebraic subvariety of $X$ containing all non constant entire curves $f\colon \mathbb{C} \to X$ . Such questions can be investigated by studying the rich geometry of jet bundles and the neativity properties of their curvature. We will try to present the main ideas of these techniques, and especially the recent proof by D.Brotbek of a long-standing conjecture of Kobayashi (1970), according to which a generic algebraic hypersurface of $\mathbb{P}^n$ of sufficiently large degree is hyperbolic in the sense of Kobayashi.

COURSE 2: Automorphisms of algebraic varieties

Igor DOLGACHEV (University of Michigan, USA)

  • General facts about the scheme of automorphisms of a projective algebraic variety
  • Action on cohomology: cohomologically trivial automorphisms
  • Automorphisms of algebraic curves and surfaces of general type
  • Automorphisms of algebraic surfaces: K3, Enriques and rational surfaces

COURSE 3: Prym varieties and Prym maps

Angela ORTEGA (Humboldt Universität, Germany)

Prym varieties are abelian varieties constructed from coverings of algebraic curves. It is well known that a general principally polarized abelian variety of dimension at most 6 is a Prym variety. For a given g, degree, and ramification degree of a covering over a genus g curve, there is a corresponding Prym map that associates to every covering of this type a polarized abelian variety. We will start with some generalities about Prym varieties; we will present the different strategies to compute the degree of the Prym map (when it is finite), and various cases of Prym maps whose fibres carry a particularly beautiful structure.

COURSE 4: Algebraic curves and automorphic forms

Vincent PILLONI (École normale supŕrieure de Lyon,France)

It has been conjectured that the zeta function of an algebraic curve defined over a number field has a meromorphic continuation to the complex plane. A strategy to prove this is to show that such curve is "automorphic". In this course, we will explain what this means.

COURSE 5: Stable birational invariants and the stable Lüroth problem

Claire VOISIN (Collège de France,France)

The lectures will discuss cohomological and Chow-theoretic obstructions to rationality or stable rationality of complex projective varieties. Of course, there are many such geometric obstructions, like the plurigenera, but we will rather focus on the case of rationally connected varieties, where these obvious obstructions are trivial. We will discuss:

  • Unramified cohomology and its link to the integral Hodge conjecture, (this provides very interesting irrationality criteria)
  • Various notions of decompositions of the diagonal

The study of the (stably birationally invariant) property of having a "decomposition of the diagonal" in the Bloch-Srinivas sense allowed Claire Voisin and subsequently other people to prove that many rationally connected varieties are not stably rational. These works culminated with the proof by Hassett-Pirutka-Tschinkel that rationality is not deformation invariant. Comparing this birational invariant with previously defined invariants (e.g. the Clemens-Griffiths criterion) is also very interesting and has been completely done in dimension 3. This could be the last point developed in the lectures, if time allows.