Schedule

All lectures will take place in the Aula Buzano (third floor) at the Department of Mathematical Sciences of the Politecnico di Torino, at Corso Duca degli Abruzzi, 24, see the map.

Short courses

Nathan Ilten

Deformation Theory of Smooth Complete Toric Varieties

There is a standard approach to understanding the infinitesimal deformation theory of a variety X: first one computes the space T¹ of first order deformations, and then the space T² of obstructions to deformation. Next, one must understand the obstruction in T² of lifting a given deformation to higher order. Starting with all first order deformations and iteratively lifting, one will arrive at a versal deformation of X.

We will study what parts of this approach looks like for a very special class of varieties: smooth complete toric varieties. In this situation, the spaces T¹ and T² are graded by a lattice, and each graded piece has a simple combinatorial description. On the one hand, we will see that it is possible to lift any homogeneous infinitesimal deformation to a flat family over affine space. On the other hand, we will see that in general there are genuine obstructions to deformation. We will do this by explicitly describing the cup product map T¹ × T¹ → T², which measures the obstruction of combining two first-order deformations.

In this course, I will assume no background in either toric geometry or deformation theory, and introduce all relevant concepts as needed. Some of the results I present will be based on joint work with Robert Vollmert and with Charles Turo.

Handwritten notes of this course.

Christian Lehn

Invariant deformation theory for reductive groups

Very often, deformation theory is used to show that a given variety is non-singular at a given point. It can however be a helpful tool beyond the question whether or not your variety is non-singular. In this series of lectures we will give an introduction to invariant deformation theory and we will explain how it can serve in practice to calculate defining equations of the variety in question up to a given finite order. We will illustrate this for the invariant Hilbert scheme of Alexeev and Brion, where in some cases it is even possible to obtain defining equations for an affine neighborhood of a given point. This method has lead to some insight about the kind of singularities the invariant Hilbert scheme can have; in particular, there are examples where the invariant Hilbert scheme is singular but its principal component is non-singular.

Mauro Porta

Derived geometry and deformation theory

This will be a series of three lectures, mainly aimed at PhD students and young postdocs. The relation between deformation theory and derived algebraic geometry is rich and interesting in both directions.

There are two main results governing this relationship: on one side, the correspondence between formal moduli problems (FMP) and differential graded Lie algebras (dgla). On the other side, Lurie's extension of Artin's representability theorem. Giving a detailed account of the technical ideas behind these theorems lies beyond the scope of this mini-course. Instead I will focus more on examples of these results, both classical and recent.

Here is a more detailed outline of the mini-course. In the first lecture I will provide examples of the classical problems that lead people to conjecture the FMP-dgla correspondence. In the second lecture I will provide a more formal background on derived algebraic geometry. Rather than insisting on the foundational aspects of the theory, I will present a series of fundamental results that make derived schemes manageable in practice. During the third lecture I will analyze in greater detail some of the fundamental examples and I will provide a couple of more advanced ones.

Talks

Francesca Carocci

Endomorphisms of the Koszul complex and deformations of lci ideal sheaves

It has been proved by Fiorenza-Iacono-Martinengo that infinitesimal deformations of a coherent sheaf are controlled by the DG-Lie algebra of endomorphisms of a locally free resolution. In the talk, we will first recall this result and then exploit the language of DG-Lie algebras to prove an annihilation result for obstructions to (derived) deformations of lci ideal sheaves. The annihilation theorem will follow from proving that DG-Lie algebra of endomorphisms of the Koszul complex for a regular sequence is homotopy abelian over the base field.

(This is based on a joint work with Marco Manetti.)

Martí Lahoz

Stability conditions in a family of non-commutative K3 surfaces

The derived category of coherent sheaves on a smooth cubic fourfold has a subcategory, recently studied by Kuznetsov, Addington-Thomas and Huybrechts among others, that can be thought as the derived category of a non-commutative K3 surface. In this talk, I will present joint work together with Bayer, Macrì, and Stellari about the construction of Bridgeland stability conditions on this category. Together with Nuer and Perry, we also study the behavior of stability conditions in families, which allows us to characterize the non-emptiness of moduli spaces of objects in this subcategory.

Andrea Petracci

Smoothing Fano toric threefolds

The deformation theory of toric singularities has been extensively studied by Klaus Altmann. Very little is known about deformations of non-smooth projective toric varieties. In this talk I will describe how to construct deformations of Fano toric threefolds with Gorenstein singularities by gluing Altmann's deformations of affine charts; in this way, it is possible to construct smoothings of (some) Fano toric threefolds with Gorenstein singularities. I will also explain the relation with Mirror Symmetry for Fano varieties.

This talk is based on ongoing collaboration with Alessio Corti and Paul Hacking.

Giorgio Scattareggia

An obstruction theory for the moduli spaces of coherent systems

Informally, a perfect obstruction theory for a moduli space M is a perfect complex in the derived category of M which encodes all the information about the infinitesimal properties of M. Every moduli space which admits a perfect obstruction theory of rank r has expected dimension r; conversely, whenever a moduli space has expected dimension r, one suspects that it has an obstruction theory of rank r.

A coherent system on a curve C is a pair (E,V), where E is a finite rank vector bundle on C and V is a linear subspace of the space of global sections of E. The type of a coherent system (E,V) is a triple (n,d,k), where n is the rank of E, d is the degree of E and k is the dimension of V. In 1998 it was proved that the moduli spaces of (semi-)stable coherent systems have an expected dimension which only depends on the genus of the curve and on the type of the coherent systems.

In this talk we briefly recall the definition of obstruction theory and we sketch the construction of a perfect obstruction theory for the moduli spaces of stable coherent systems.

Sinan Yalin

Derived Etingof-Kazhdan deformation quantization and higher Hochschild cohomology

I will present a work in collaboration with Gregory Ginot which solves and generalize at once a host of longstanding conjectures in deformation theory and deformation quantization, by setting them in an appropriate new framework. I will first introduce this framework to parametrize the homotopy theory of a large class of algebraic structures, as well as the derived formal moduli problems controlling their deformation theory. Relying on this solid basis, I will explain why the deformation theory of conilpotent dg bialgebras is controled by the higher Hochschild complex of algebras over the little 2-disks operad.

By the higher Deligne conjecture (now a theorem), this higher Hochschild complex inherits a structure of algebra over the little 3-disks operad, and it turns out that this structure controls the deformations of bialgebra structures, hence solving a longstanding conjecture of Gerstenhaber-Schack (1990). This result, in turn, can be used to prove an E3-formality theorem for the deformation complex of the symmetric dg bialgebra conjectured by Kontsevich in his work on deformation quantization of Poisson manifolds (2000). As a consequence this gives a new proof of Etingof-Kazdhan's quantization theorem for Lie bialgebras, parallel to Kontsevich-Tamarkin's proof of deformation quantization of Poisson manifolds, and generalized to the setting of homotopy dg Lie bialgebras. More generally, we define an infinity-functor appropriately generalizing the Etingof-Kazdhan quantization functor and which can be seen as a "Kontsevich formality result in families".