Schedule for: 20w5176 - Derived, Birational, and Categorical Algebraic Geometry (Online)

A brief introduction video from BIRS staff with important logistical information

In a joint work with Yu.Prokhorov we established rationality criteria for geometrically rational Fano threefolds over non-closed fields of characteristic zero such that their geometric Picard number is one. I will report on similar results for Fano threefolds whose geometric Picard number is higher but the Picard number over the base field is one. I will also describe the derived categories of these varieties over the base field and discuss the relation between their structure and rationality properties.

Consider a diagrammatic category whose objects are partitions of n and whose morphisms are braids with multiplicities where strands are allowed to merge and come apart, so topologically such a braid is a trivalent graph with boundary. In addition, we add framing on edges with multiplicities greater than 1. The usual (type A) braid group is then the group of automorphisms of (1,1,...,1). We prove that any DG enhanceable triangulated category D with a braid group action (of which there are numerous examples in algebraic geometry) can be completed to a representation of this diagrammatic category. We do this by constructing a monad over D that is best described as the nil Hecke algebra generated by the generators of the braid group action, and considering suitable categories of modules over its "block subalgebras". If D=D(X), those modules would be complexes of sheaves on X with additional data. Similar structures have been known before, but they satisfy stronger conditions (i.e. the twist of framing on a multiple strand being a shift, which in our construction is not the case). This is joint work in progress with Timothy Logvinenko.

I will report on joint work with Jenia Tevelev answering a question of Orlov. We prove that the Grothendieck-Knudsen moduli spaces of pointed stable rational curves with n markings admit full, exceptional collections which are invariant under the action of the symmetric group $S_n$ permuting the markings. In particular, a consequence is that the K-group with integer coefficients is a permutation $S_n$-lattice.

I will explain that stability conditions for general Gorenstein terminal 3-fold flops can be described as a covering map over something reasonable. Basically, part of the description comes from the movable cone, and its image under tensoring by line bundles. Alas, there is more. This extra stuff is not immediately "birational" information, and it is a bit mysterious, but it does have a very natural noncommutative interpretation, with geometric corollaries. In the process of this, I'll describe some of the new hyperplane arrangements that arise, which visually are very beautiful. If time allows, I will also explain some applications to autoequivalences and to curve counting. This is joint work with Yuki Hirano, and with Osamu Iyama.

In this talk, I will discuss the problem of computing the cohomology of the general sheaf in a moduli space of sheaves on a surface. I will concentrate on the case of rational and K3 surfaces. The case of rational surfaces uses the stack of prioritary sheaves and is joint work with Jack Huizenga. The case of K3 surfaces uses Bridgeland stability and is joint work with Howard Nuer and Kota Yoshioka.

Fourier-Mukai functors are central to the study of derived categories, but they don't tell the whole story: in 2014 we constructed the first functor between the derived categories of two smooth projective varieties that is not Fourier-Mukai. In this talk I will show that non-Fourier-Mukai functors are actually quite common and not a pathological occurrence. This is joint work with Theo Raedschelders and Michel Van den Bergh.

The homological interpretation of the Minimal Model Program conjectures that flips should correspond to embeddings of derived categories, and flops to equivalences. Even if the conjecture doesn’t provide us with a preferred functor, there is an obvious choice: the pull-push via the fibre product. When this approach work, we obtain an interesting autoequivalence of either side of the flop, known as the “flop-flop autoequivalence”. Understanding the structure of this functor (e.g. does it split as the composition of simpler functors?) is an interesting problem, and it has been extensively studied. In this talk I will explain that there is a natural, i.e. arising from the geometry, way to realise the “flop-flop autoequivalence” as the inverse of a spherical twist, and that this presentation can help us shed light on the structure of the autoequivalence itself.

An involution of a projective hyperkähler manifold is called antisymplectic if it acts as (-1) on the space of global holomorphic 2-forms. I will present joint work with Laure Flapan, Kieran O'Grady, and Giulia Saccà on antisymplectic involutions associated to polarizations of degree 2. We study the number of connected components of the fixed loci and their geometry.

Hochschild cohomology behaves well over a field, and its derived analogue Shukla cohomology behaves well over any base commutative ring. Both are intimately related to deformation theory. To study `nonlinear' deformations (e.g. Z/p^2 over Z/p), one wants to study Mac Lane cohomology, which introduces nonadditive features. Mac Lane cohomology ought to be the same thing as topological Hochschild cohomology; the analogue for homology is known by work of Pirashvili and Waldhausen. I'll give a quick recap on topological Hochschild cohomology, which is morally just Shukla cohomology with base `ring' the sphere spectrum. I'll then give a definition of THH^* for schemes, along with some comparison theorems showing that for reasonable schemes, any of the `obvious' definitions that one might make all agree. I'll give some (easy!) computations of THH^* for P^1 and P^2 over a finite field.

We study certain sequences of moduli spaces of sheaves on K3 surfaces, building on work of Markman, Yoshioka, and Nakajima. We show that these sequences can be given the structure of a geometric categorical sl_2 action in the sense of Cautis, Kamnitzer, and Licata. As a corollary, we get an equivalence between derived categories of some moduli spaces that are birational via stratified Mukai flops. I'll spend most of my time on a nice example. This is joint with my student Ryan Takahashi.

Using recent advances in the Minimal Model Program for moduli spaces of sheaves on the projective plane, we compute the cohomology of the tensor product of general semistable bundles on the projective plane. More precisely, let V and W be two general stable bundles, and suppose the numerical invariants of W are sufficiently divisible. We fully compute the cohomology of the tensor product of V and W. In particular, we show that if W is exceptional, then the tensor product of V and W has at most one nonzero cohomology group determined by the slope and the Euler characteristic, generalizing foundational results of Drézet, Göttsche and Hirschowitz. We also characterize when the tensor product of V and W is globally generated. Crucially, our computation is canonical given the birational geometry of the moduli space, providing a roadmap for tackling analogous problems on other surfaces. This is joint work with Izzet Coskun and John Kopper.

Bridgeland stability manifolds of Calabi-Yau categories are of noticeable interest both in mathematics and in physics. By looking at some of the known example, a pattern clearly emerges and gives a fairly precise description of how they look like. In particular, they all seem to have missing loci, which tend to correspond to degenerate stability conditions vanishing on spherical objects. Describing such missing strata is also interesting from a mirror-symmetric perspective, as they conjecturally parametrize interesting types of degenerations of complex structures. All the naive attempts at constructing modular partial compactifications show how elusive and subtle the problem in fact is: ideally, the missing strata would correspond to stability manifolds of quotient triangulated categories, but establishing such correspondence on geometric level and viewing stability conditions on quotients of the original triangulated category as suitable degenerations of stability conditions is not straightforward. In this talk, I will present method to construct such partial compactifications if some additional hypoteses are satisfied, by realizing our space of interest as a suitable metric completion of the stability manifold.

After an overview of Hochschild homology and topological Hochschild homology, I will talk about about the twisted versions of these that can be defined in the presence of an action of a finite cyclic group. I will discuss joint work with Adamyk, Gerhardt, Hess, and Kong in which we develop a theoretical framework and computational tools for these twisted Hochschild homology theories.

We study the behavior of the pullback map between the Brauer group of a bielliptic surface and that of its canonical cover. This is joint work with E. Ferrari and. Vodrup.