Schedule

The workshop will take place in Aula Monod on the second floor of the Department of Mathematics (Palazzo Campana) of the University of Turin.

Talks

Donatella Iacono: Deformations of varieties and vector bundles

In this talk, we focus our attention on the obstructions to infinitesimal deformations of pairs (variety, vector bundle), using differential graded Lie algebras. In particular, we prove the unobstruction of deformations of pairs (variety with trivial canonical bundle, line bundle). Joint work in progress with Marco Manetti.

Andreas Krug: Semi-orthogonal decompositions on Hilbert schemes of points

I will survey results on semi-orthogonal decompositions on Hilbert schemes of points from joint work with Pawel Sosna and joint work with Pieter Belmans. Some of the results are applications of a recent interesting result by Jiang and and Leung on the derived category of the projectivisation of a coherent sheaf that we will also discuss.

Francesco Meazzini: On the Kaledin-Lehn formality conjecture

Kaledin-Lehn conjectured the formality for the DG-Lie algebra of derived endomorphisms of any polystable sheaf on a K3 surface. The relevance of the formality conjecture relies in its consequences concerning the geometry of the moduli space of semistable sheaves on the K3. The conjecture was proven after several contributions mainly due to Kaledin-Lehn themselves, Zhang, Yoshioka, Arbarello-Saccà, Budur-Zhang. We propose an alternative algebraic approach to the problem, eventually proving that the formality conjecture holds on smooth minimal projective surfaces of Kodaira dimension 0. This is a joint work with R. Bandiera and M. Manetti.

Valerio Melani: Weinstein's "Poisson category" in derived algebraic geometry

Motivated by deformation quantization, Weinstein initiated the study of the "Poisson category". This should be a category whose objects are Poisson manifolds, and whose morphisms are coisotropic correspondences. Unfortunately, in the general case there is no such category. In fact, composition of morphisms by fiber products is not always available, and one needs to put strong enough "clean intersection" hypothesis to make it possible. In this talk, we present a realization of the Poisson category in the context of derived algebraic geometry, which is a homotopical generalization of classical algebraic geometry. The talk will be based on joint work(s) with Rune Haugseng and Pavel Safronov.

David Ploog: A graphical approach to cohomology of line bundles on toric varieties

Line bundles on smooth projective toric varieties have a polytopic description, and hence so has their cohomology. This has been used to investigate full exceptional sequences of line bundles on such varieties. In this talk, we state and prove a formula for line bundle cohomology using the M-lattice. (Joint with Klaus Altmann.)

Mattia Talpo: Towards logarithmic degeneration for DT invariants

Donaldson-Thomas invariants count algebraic curves on Calabi-Yau 3-folds via counting their sheaves of ideals. As on the Gromov-Witten side, there are techniques to reduce computations to simpler geometries, by degenerating the manifold to a certain kind of singular varieties. Logarithmic geometry is a modern language that helps in handling these kinds of degenerations, and is being fruitfully applied to degeneration of GW invariants. After some review of these topics, I will speculate about possible applications of these new techniques to degeneration of DT invariants.

Fabio Tonini: Representations of the Nori fundamental group and variants

The Nori fundamental group scheme of a scheme X is a profinite group scheme that "controls" torsors over X under finite group schemes. In the talk I will describe its category of representations by considering vector bundles with extra structure. I will also talk about a variant of this group, the local fundamental group scheme.

Davide Veniani: The two most algebraic Enriques surfaces

In 2007 Ohashi observed that any K3 surface covers a finite number n of Enriques surfaces up to isomorphism. Building on his work, we gave an exact formula for this number n in terms of lattice theory. Together with Shimada, we computed n for all singular K3 surfaces of discriminant less than or equal to 36. We discovered that the K3 surface of discriminant 7 — the smallest one for which n > 0 — covers exactly two Enriques surfaces: the two most algebraic ones. We investigate their automorphism groups and give explicit equations. If time permits, I will also report on an ongoing project with Brandhorst and Sonel on the classification of K3 surfaces which do not cover any Enriques surface.