Journées GAGC à Angers

Gromov--Witten Theory of Complete Intersections

I will explain an inductive algorithm for computing Gromov--Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. For this, I will first discuss how to trade invariants with primitive insertions (insertions that are not pulled back from the projective space) against nodal invariants with non-primitive insertions. Then, to compute such nodal invariants I will state nodal degeneration and splitting formulas. This is joint work with Pierrick Bousseau, Rahul Pandharipande, and Dimitri Zvonkine.

Birational involutions on Hilbert schemes of points on a K3 surface

In a joint work with Alberto Cattaneo, we studied the group of birational automorphisms of Hilbert schemes of n points on a K3 surface, which are an important example of hyperkäler manifolds in dimension 2n. A very interesting result is that there exists birational involutions acting trivially on a 2-form on the Hilbert scheme, even when the latter has minimal Picard rank, while in the biregular case this cannot happen. Many results can be deduced by the study of the linear system associated to a divisor whose class generates the invariant lattice of an involution: we will present some results in dimension 4 and 6.

Blown up toric surfaces with non-polyhedral effective cone

I will discuss joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia on constructing examples of projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone, both in characteristic 0 and in every prime characteristic p. As a consequence, we prove that the pseudo-effective cone of the Grothendieck-Knudsen moduli space of stable rational curves with n markings is not polyhedral for n>=10. Many of these toric surfaces are related to an interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order.

Characterizing quotients of complex tori

In a joint work with Patrick Graf and Henri Guenancia we studied a singular analogue of Yau's theorem saying that a compact Kähler manifold whose first and second Chern classes vanish admit an étale cover by a complex torus. To generalize this kind of result to the klt case, we establish a singular version of the Bogomolov-Gieseker inequality. Our proof finally relies on the recent Decomposition Theorem for Kähler Ricci flat spaces (a result of Bakker--Guenancia-Lehn).

What are the mixed Tate motives of rank 2?

The category of mixed Tate motives over a number field (iterated extensions of the Tate motives Q(-n) for all integers n) is well understood from an abstract viewpoint, but the structure of its periods is still largely mysterious. This talk will be about the explicit description of rank 2 mixed Tate motives, i.e., extensions of Q(-n) by Q(0), a question which is motivated by the understanding of the special values of Dedekind zeta functions. I will explain how this also sheds light on the search for irrationality proofs for certain mathematical constants.

Applications of geometric generalized Wronskians in hyperbolicity

During this talk, we will recall the definition of generalized Wronskians, and exhibit a sub-family, whose elements are called geometric. Those geo- metric generalized Wronskians have two advantages: on the one hand, they allow global geometric constructions, that we will describe, and on the other hand, they still allow to detect linear independance of holomorphic functions (which is the fundamental property of generalized Wronskians, known since the work of Roth in ∼ 1950). We will then present an application of this construction in hyperbolic- ity, and more precisely in the study of family of entire curves in Fermat hypersurfaces. If time allows, we will also discuss applications to foliations.

On the moduli part of a klt-trivial fibration

Let f: (X,B)->Y be a fibration such that the log-canonical divisor K+B is trivial along the fibres of f. The canonical bundle formula is a way of expressing K+B as the pullback of the sum of three divisors: the canonical divisor on Y; a divisor, called discriminant, carrying informations on the singular fibres; a divisor called moduli part keeping track of the birational variation of the fibres and having positivity properties.
Let T be a connected, reduced but not necessarily irreducible divisor of Y.
In this talk I will give a geometric interpretation of the semiampleness of the restriction of the moduli part to T and give some results towards the semiampleness of the restriction.

Derived intersections and Lie algebroids

In this talk, we will explain (after the work of Calaque, Caldararu and Tu) how the formal geometry of a smooth subscheme can be encoded in a derived Lie algebroid. Then we will present different geometric situations where this Lie algebroid can be described explicitly. Joint work with Damien Calaque.

Deformations of solutions of differential equations

Given an algebraic differential equation (or system) we can be interested in the geometrical object defined by the set of its solutions. In algebraic geometry, the study of the geometry of the solutions of a polynomial equation gives information about the singularities of this equation. If we transpose some of these algebraic invariants into the differential world, we can wonder what information they give about the differential equation. In this presentation, we will study the behavior of the formal neighborhood of a fixed solution of the differential equation. In our framework, this object can also be understood as the space of deformations of this solution.

Real forms of some classes of almost homogeneous varieties

Let G be a complex reductive algebraic group, and let X be a normal G-variety. In this talk, we will discuss the notion of real structures on X which are compatible with a given real structure on G. In the 1980’s Luna and Vust gave a combinatorial description of almost homogeneous complex varieties. It is particularly effective in the case of complexity zero or one. We will show how to use this data to determine the real structures of these G-varieties. For the purposes of this talk, we will concentrate on the case of three-dimensional almost homogeneous SL_2-varieties. We will also talk about some other cases. Finally, we will discuss how these results generalize to the case of G-varieties over a general perfect field. This is joint work with Ronan Terpereau.