Organisateurs : Benoît Bertrand, Erwan Brugallé, Ilia Itenberg, Grigory Mikhalkin

Toulouse, 6 novembre 2014

Institut de mathématiques de Toulouse, Salle 207

10h:30 Coffee

11h Andrei Teleman (Université de Marseille)

Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces

We explain a method to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. In particular we describe explicitly the group of isomorphism classes Real line bundles on a Real torus. On a Klein surface the determinant index bundles we study have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control the orientability of certain moduli spaces in Real gauge theory and Real algebraic geometry. The results is based on combination of algebraic geometric, topological and differential geometric arguments.

14:00 Thomas Dedieu (Université de Toulouse 3)

Limits of pluri-tangent planes to quartic K3 surfaces

I will discuss the following general question : given a family X → D of compact complex surfaces in P N with smooth general member (here D stands for the complex unit disk), what are the limits as t ∈ D tends to 0 of the families of hyperplanes tangent in d points to X t, t ≠ 0 ? (here d is a given integer, smaller than N) I will give a complete answer in the two cases of smooth quartic hypersufaces in P 3 degenerating to (i) the union of four planes, and (ii) a Kummer surface. I shall then sketch a conjectural answer to the question in general. An application to the study of the irreducibility and enumerative geometry of the Severi varieties of quartic K3 surfaces will eventually be given. This is joint work with Ciro Ciliberto.

15h coffee

15h30 Tony Yue Yu (Paris 7)

Counting cylinders via non-archimedean geometry

Counting the number of curves in an algebraic variety is a classical topic in algebraic geometry. Inspired by string theory and mirror symmetry, people started to look at not only closed curves, but also discs, cylinders, etc. An obvious difficulty of counting discs or cylinders is that they are no longer objects in algebraic geometry. Following the suggestions by Kontsevich and Soibelman in 2000, we use non-archimedean geometry to tackle the problem. I will begin by reviewing non-archimedean techniques in tropical geometry and some of my related previous results. Then I will explain the case of log Calabi-Yau surfaces in detail.