Smooth, complex Fano 4-folds are not classified, and we still lack a good understanding of their general properties. We focus on Fano 4-folds with large second Betti number $b_{2}$, studied via birational geometry and the detailed analysis of their contractions and rational contractions (we recall that a contraction is a morphism with connected fibers onto a normal projective variety, and a rational contraction is given by a sequence of flips followed by a contraction). The main result that we want to present is the following: let $X$ be a Fano 4-fold having a nonconstant rational contraction $X --> Y$ of fiber type. Then either $b_{2}(X)$ is at most 18, with equality only for a product of surfaces, or $Y$ is $\mathbb{P}^{1}$ or $\mathbb{P}^{2}$. The proof is achieved by reducing to the case of "special" rational contractions of fiber type. We will explain this notion and give an idea of the techniques that are used.[-]

Gushel-Mukai varieties are defined as the intersection of the Grassmannian Gr(2, 5) in its Plücker embedding, with a quadric and a linear space. They occur in dimension 6 (with a slighty modified construction), 5, 4, 3, 2 (where they are just K3 surfaces of degree 10), and 1 (where they are just genus 6 curves). Their theory parallels that of another important class of Fano varieties, cubic fourfolds, with many common features such as the presence of a canonically attached hyperkähler fourfold: the variety of lines for a cubic is replaced here with a double EPW sextic.
There is a big difference though: in dimension at least 3, GM varieties attached to a given EPW sextic form a family of positive dimension. However, we prove that the Hodge structure of any of these GM varieties can be reconstructed from that of the EPW sextic or of an associated surface of general type, depending on the parity of the dimension (for cubic fourfolds, the corresponding statement was proved in 1985 by Beauville and Donagi). This is joint work with Alexander Kuznetsov.[-]

In the talk I will discuss rationality criteria for Fano 3-folds of geometric Picard number 1 over a non-closed field $k$ of characteristic 0. Among these there are 8 types of geometrically rational varieties. We prove that in one of these cases any variety of this type is k-rational, in four cases the criterion of rationality is the existence of a $k$-rational point, and in the last three cases the criterion is the existence of a $k$-rational point and a k rational curve of genus 0 and degree 1, 2, and 3 respectively. The last result is based on recent results of Benoist-Wittenberg. This is a joint work with Yuri Prokhorov.[-]

It is thought that the classifications and constructions of holomorphic Poisson structures are worth studying. The classification when the Picard rank is 2 or higher is unknown. In this talk, we will introduce the classification of holomorphic Poisson structures with the reduced degeneracy divisor that have only simple normal crossing singularities, on the product of Fano variety of Picard rank 1. This claims that such a Poisson manifold must be a diagonal Poisson structure on the product of projective spaces, so this is a generalization of Lima and Pereira's study. The talk will also include various examples, classifications, and problems of high-dimensional holomorphic Poisson structures.[-]

In diophantine geometry, the Batyrev-Manin-Peyre conjecture originally concerns rational points on Fano varieties. It describes the asymptotic behaviour of the number of rational points of bounded height, when the bound becomes arbitrary large.
A geometric analogue of this conjecture deals with the asymptotic behaviour of the moduli space of rational curves on a complex Fano variety, when the 'degree' of the curves 'goes to infinity'. Various examples support the claim that, after renormalisation in a relevant ring of motivic integration, the class of this moduli space may converge to a constant which has an interpretation as a motivic Euler product.
In this talk, we will state this motivic version of the Batyrev-Manin-Peyre conjecture and give some examples for which it is known to hold : projective space, more generally toric varieties, and equivariant compactifications of vector spaces. In a second part we will introduce the notion of equidistribution of curves and show that it opens a path to new types of results.[-]

Positivity of the tangent bundle and its top exterior power (namely, the anticanonical bundle) is the subject of extensive literature, and several open problems. Notably, Campana and Peternell predict that if $-K_{X}$ is strictly nef, then $X$ is a Fano variety: this conjecture is proven up to dimension 3 by the work of Maeda and Serrano. In this talk, we investigate positivity of the intermediate exterior powers of the tangent bundle. We prove that if $X$ is a smooth projective n-fold and the third, fourth or (n-1)-th exterior power of $T_{X}$ is strictly nef, then $X$ is a Fano variety. Moreover, we classify smooth projective varieties of Picard number at least two with third or fourth exterior power of $T_{X}$ strictly nef. This work is actually slightly more general, as it boils down to classifying rationally connected varieties such that the degree of the anticanonical bundle on rational curves is quite large.[-]

I will present a joint work with Sara Angela Filippini, Laurent Manivel and Fabio Tanturri (arXiv: 1704.01436). We introduce a new class of varieties, called orbital degeneracy loci. The idea is to use any orbit closure in a representation of an algebraic group to generalise the classical construction of degeneracy loci of morphisms between vector bundles, and of zero loci as well. After giving the definition of an orbital degeneracy locus, I will explain how to control the canonical bundle of these varieties: under some Gorenstein condition on the orbit closure, it is possible to construct examples of varieties with trivial canonical bundle or of Fano type. Finally, if time will permit, I will give some explicit examples of such degeneracy loci, which allow to construct many Calabi-Yau varieties of dimension three and four, and some new Fano fourfolds.[-]