Workshop
“Derived categories and birational geometry”

Consider a contraction from a smooth Calabi-Yau 3-fold to a singular one. (This is half of an "extremal transition;" the other half would be a smoothing of .) In many examples there is an intermediate object called an "exoflop" -- a category of matrix factorizations, derived-equivalent to , where the critical locus of the superpotential looks like with a sticking out of the singularity, and objects of that will be killed by correspond to objects supported at the far end of the . I will discuss several interesting examples. This is joint work with Paul Aspinwall.

In a joint work with Alan Thompson, we describe a modular compactification of the moduli space of polarized K3 surfaces of degree 2. I will also provide some speculations relating this compactification to a construction of Kontsevich-Soibelman from homological mirror symmetry.

This is a joint work in progress with A.Auel. Let S be a minimal geometrically rational surface with ample anticanonical bundle. We study the orthogonal complement A of the canonical bundle in the derived category of S, and show how birational informations on S can be recovered from it. For example, S is rational if and only if A is representable in dimension 0. If the degree of S is at least 5, semiorthogonal decompositions of A give rise to a pair of vector bundles with semisimple endomorphism algebras whose classes in the Brauer groups of the respective centers give a birational invariant of S. Finally, the index of S can be calculated both from the index of these algebras and from the second Chern classes of those vector bundles.

For a flopping contraction of Gorenstein three-folds Bridgeland introduced the family of t-structures labeled by an integer parameter p. Hearts of t-structures with perversity p and q where shown to be equivalent when p is congruent to q modulo 2. In my talk I show that t-structures for perversity p and p+1 are related by a tilt in a torsion pair. I also prove that the equivalence of hearts is given by a spherical twist on the derived category of the variety. The associated spherical functor arises from the canonical embedding of the category of coherent sheaves with vanishing derived direct image. This is a joint work with Alexey Bondal.

We discuss various presentations for derived equivalences of flops of relative dimension 1. We emphasise the importance of the null-category and give two presentations of the flop-flop functors via spherical twists. The key technical ingredient is the lemma of L^1-vanishing, which is of independent interest. This is a joint work with Agnieszka Bodzenta.

TBA

I will survey on a conjecture due to Shokurov on the ACC for the set of minimal log discrepancies and I will describe an approach towards this conjecture using toroidal modifications. Joint work with J. McKernan.

I will discuss a version of the Sarkisov program for volume-preserving maps of Mori fibred Calabi-Yau pairs. This is a joint work with A-S. Kaloghiros.

A terminal divisorial extraction from a curve C in a 3-fold X (if it exists) is given by the blowup of the symbolic power algebra of the ideal defining C. For X smooth, I will explain how you can construct these rings explicitly using unprojection. In particular, if C is contained in a hypersurface with Du Val singularity of type A this leads to large families of graded rings with an interesting and complicated combinatorial structure.

Kontsevich and Soibelman formulated a conjecture on degeneration of Hochschild-to-cyclic spectral sequence for saturated DG categories over a field of characteristic zero. In this talk I will propose a more general conjecture which states identical vanishing of a certain map between bi-additive invariants of an arbitrary pair of small DG categories (again, over a field of characteristic zero). It turns out that this general conjecture is implied by Kontsevich-Soibelman conjecture and a certain (open) conjecture on "smooth categorical compactification".

It turns out that our conjecture is non-trivial already for associative algebras. We are able to prove it in this case only over a field of positive characteristic, and I will sketch a proof.

Recent results have revealed a mysterious foundational phenomenon: some quotient stacks behave as if they are proper schemes, even when is a positive dimensional algebraic group and itself is not proper. For instance, one can consider the non-commutative Hodge-to-de-Rham sequence, from Hochschild homology to periodic cyclic homology, for the dg-category . This spectral sequence degenerates on the first page for smooth and proper schemes, and it turns out that this degeneration also occurs for many "cohomologically proper" quotient stacks. With a little work, this leads to a canonical weight 0 Hodge structure on the Atiyah-Segal equivariant K-theory of the complex analytification of . The associated graded of the Hodge filtration is the space of functions on the "derived loop space" of the stack.

Smooth cubic fourfolds are linked to K3 surfaces via their Hodge structures, due to work of Hassett, and via Kuznetsov's K3 category A. The relation between these two viewpoints has recently been elucidated by Addington and Thomas.

In the talk I shall explain how both aspects can be extended to twisted K3 surfaces, which in particular allows us to determine the group of autoequivalences of A for the general cubic fourfold. Furthermore, we prove finiteness results for cubics with equivalent K3 categories and study periods of cubics in terms of generalized K3 surfaces.

We extend results of Chan and Ingalls concerning the minimal model program for orders over surfaces to all dimensions. A decoration gives a number for all divisors of all models of a variety. We show that every decorated variety has a terminal resolution. We further show that if one carries out log contractions then decorated terminal varieties remain decorated terminal. As an application, one obtains a decoration from a Brauer class and that this can be used to give a minimal model program for orders over varieties in all dimensions.

This work is the joint work of: Daniel Chan, Kenneth Chan, Louis de Thanho er de Volcsey, Colin Ingalls, Kelly Jabbusch, Sandor Kovacs, Rajesh Kulkarni, Boris Lerner and Basil Nanayakkara.

The category of derived Mackey functors is a certain triangulated category associated to a finite group ; derived Mackey profunctors are generalizations of Mackey functors to infinite groups. The category is algebraic in the sense of Keller — that is, it has a DG-enhancement. This enhancement can be obtained by gluing simpler categories, but in a rather non-trivial and interesting way. While at present, all this has nothing to do with categorical study of algebraic varieties, it seems that the homological phenomena exhibited by Mackey functors and profunctors should appear in many other contexts including the geometric one, so it makes sense to advertise the theory. This is what I am going to do.

We will consider a examples of sheaves of categories Applications to HMS and stable nonrationality will be discussed.

Let be a finite subgroup of . McKay found a correspondence between:

1. The quiver consisting of irreducible representations of that is defined using the tensor product with the natural representation.
2. The extended Dynkin diagram of exceptional curves on the minimal resolution of the quotient space .

This phenomena is understood as an equivalence of derived categories between the quotient stack and . Such equivalence is extended to the case of a finite subgroup of by Bridgeland-King-Reid.

In this talk we consider the case where is a finite abelian subgroup of . The quotient space . is a toric variety and there exists a toric terminalization . We explain how to use the toric minimal model program to prove the following theorem; there exists a semi-orthogonal decomposition of the derived category of the quotient stack . into the derived category of and other derived categories of smaller dimensional quotient stacks.

A recent theorem of Daniel Bergh is an analogue for DM stacks of the weak factorization theorem for algebraic varieties. We will discuss two applications: 1) Geometricity of derived categories of smooth DM stacks; 2) Comparison of equivariant and usual categorical motivic measures. Some conjectures of Galkin and Shinder.

We are going to discuss different properties of noncommutative schemes. Gluing of noncommutative schemes will be defined. We introduce a notion of geometric noncommutative schemes and using birational geometry we show that the world of all geometric noncommutative schemes is closed under gluing. As a consequence it is shown that for any finite dimensional algebra the category of perfect complexes over it is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme. We provide different constructions of smooth projective schemes that admit a full exceptional collection and contain as a subcollection an exceptional collection given in advance. We also construct strong exceptional collections of vector bundles on smooth projective varieties that have a prescribed endomorphism algebra. It is proved the construction problem always has a solution. Some applications to noncommutative projective planes and to the quiver connected with the 3-point Ising function will be considered.

Firstly, I will apologise: I gave this talk in Warwick 2 months ago, but I am not so prolific as to have another ready in time. This one will be 10 times more polished, but more-or-less the same.

Secondly, I will review Beilinson's theorem, Orlov's family version, and then how this leads naturally to Kuznetsov's wonderful theory of Homological projective duality. Basically I am advocating reading his paper backwards to make the theory easier to understand (for me).

HPD is a generalisation of classical projective duality that relates derived categories of coherent sheaves on different algebraic varieties. I will explain some applications that occur in joint work with Addington, Calabrese and Segal.

I will talk about a work in progress on the constructions of Kapranov’s NC-scheme structures on the moduli spaces of stable sheaves on algebraic varieties which represent the non-commutative deformation functors of stable sheaves. Our result generalizes the known constructions of non-commutative moduli spaces of sheaves studied in specific situations by Kapranov, Segal, Polishchuk-Tu and Donovan-Wemyss. In particular, this work has possible applications to generalize the work of Donovan-Wemyss on the constructions of twist functors along floppable (1, -3)-curves on 3-folds via non-commutative deformations. If time permits, I will also discuss enumerative geometry using the NC-scheme structures on the above moduli spaces.

This is joint work with Špela Špenko. We generalize standard results about non-commutative resolutions of finite groups to arbitrary reductive groups. We show in particular that quotient singularities for reductive groups always have non-commutative resolutions in an appropriate sense. Moreover we exhibit a large class of such singularities which have (twisted) non-commutative crepant resolutions (NCCRs).

We discuss a number of examples, both new and old, that can be treated using our methods. Notably we prove that twisted NC(C)Rs exists in previously unknown cases for determinantal varieties of symmetric and skew-symmetric matrices.

I will explain a little about the homological version of the minimal model program, which interprets the flop functor equivalence of Bridgeland-Chen as a mutation functor in cluster theory. This gives a surprising number of corollaries and new results, and in the first half I will outline some of them. In the second half I will describe an application to the braiding of flops in dimension three, and their combinatorics. Unexpectedly, the braiding of these derived functors is controlled not by the standard braid group of a Dynkin diagram, or even the braid group of a Coxeter group, but instead by a naturally occurring hyperplane arrangement. I will give some examples, and explain that to understand autoequivalences of smooth 3-folds, we are forced to first understand derived equivalences between singular surfaces. The second half of the talk is joint work with Will Donovan (1504.05320).

Log resolution and simple normal crossing divisors play important role when studying singularities. We show existence of strong canonical Hironaka desingularization with normal crossings fibers. In the case of morphism the desingularization resolves all the fibers so they become simple normal crossing varieties. This allows to study the invariants of the singularities of fibers of morphisms and families of proper varieties. In particular we study the dual complexes and weight filtration on the cohomology of proper complex varieties and obtain natural upper bounds on its size.