Program

Ekaterina Amerik: Isotropic boundary of the ample cone

Joint work with M. Verbitsky and A. Soldatenkov. It follows from an old result of Kovacs that the ample cone of a projective K3 surface of Picard number at least three is either "round" (equal to the positive cone), or has "no round part", that is the boundary of the ample cone is nowhere dense in the isotropic cone. With a help of hyperbolic geometry, this easily generalizes to the hyperkähler case. We pursue this a bit further and prove that any real analytic curve in the projectivization of the isotropic boundary of the ample cone lies in a sphere contained in this isotropic boundary. We discuss how the union of maximal such spheres, the ``Apollonian carpet'', equal to the union of all analytic curves on the projectivized isotropic boundary, can look like.

Michele Bolognesi: Odd determinant moduli spaces of vector bundles on a genus 2 curve

Let C be a genus two curve. The moduli space SU_C(3) of rank three semistable vector bundles on C with trivial determinant a double cover of P8 branched over a sextic hypersurface, whose projective dual is the famous Coble cubic, the unique cubic hypersurface that is singular along the Jacobian of C. Let V be a 9-dimensional complex vector space. Starting from a general trivector v in wedge^3(V), I will construct a Fano manifold D_{Z_10}(v) inside the Grassmannian G(3,9) as an orbital degeneracy locus. It turns out that D_{Z_10} naturally defines a family of Hecke lines in SU_C(3). With some work, this property allows us to deduce that D_{Z_10}(v) is isomorphic to the odd moduli space SU_C(3,O_C(c)) of rank three stable vector bundles on C with fixed effective determinant of degree one. As a side result, I will show that the intersection of D_{Z_10}(v) with a general translate of G(3,7) inside G(3,9) is a K3 surface of genus 19. This is joint work with V. Benedetti, D. Faenzi and L. Manivel.

Bruno Dewer:  A characterization of quadric bundles in relative dimension 1 and 2

In 2017 A.Höring and C.Novelli proved that an elementary Mori contraction of fibre type and maximal length is birational to a projective bundle. Assume now that the length is submaximal. We construct a birational model which is equidimensional and has a smooth base. We suspect that this model is a quadric bundle, which is proven in codimension one when the relative dimension is 1 or 2.

Lie Fu: Hochschild-Serre cohomology and Hilbert schemes of points on surfaces

We explore the deformation theory of the Hilbert scheme of points on surfaces using non-commutative methods, specifically through the computation of their Hochschild cohomology. This analysis reveals that the computation requires placing ourselves in the broader framework of Hochschild-Serre cohomology—a bigraded cohomology theory that extends both Hochschild cohomology and Hochschild homology, and in the more general setting of symmetric quotient stacks of arbitrary dimension.

Through this approach, we derive a range of consequences, encompassing and unifying results of Boissière, Fantechi, Hitchin etc, pertaining to the deformation theory of the Hilbert schemes of points on surfaces. This is based on a joint work with Pieter Belmans and Andreas Krug, arXiv:2309.06244.

Crislaine Kuster: Foliations on homogeneous varieties

Let X be a projective homogeneous variety, i.e. varieties which admit a Lie group acting on it transitively. The projective space and Grassmannians are examples of such varieties. Consider an embedding of X in a projective space P^n. In this short-talk, I will present a survey on the theory of foliations on homogeneous varieties. In particular, we will be interested in whether a foliation on X is a restriction of a foliation on the ambient space P^n or not.

Erwan Rousseau: An Albanese construction for Campana’s C-pairs

We will explain a construction of Albanese maps for orbifolds (or C-pairs) with applications such as a generalization of the Bloch-Ochiai theorem. (Joint with Stefan Kebekus).

Andrea Fanelli and Nicolas Perrin: Rational simple connectedness

In this series of talks we present work of Deland on rationally simply connected varieties, which were introduced by de Jong and Starr in connection with generalizations of the Tsen-Lang theorem. The main application is a proof that 2-Fano complete intersections are rationally simply connected by constructing 2-twisting surfaces.