THE GEOMETRY OF DERIVED CATEGORIES
University of Liverpool, September 9-13, 2019

Indecomposability of symmetric powers of curves
by Pieter Belmans

Abstract: It is expected that $$\operatorname{Sym}^iC$$ has an indecomposable derived category, provided that $$i \leq g-1$$. This is now known for $$i$$ up to the gonality of the curve. By introducing a deformation theory for semiorthogonal decompositions we can improve this, and obtain indecomposability for all $$i$$ up to roughly $$g/2$$. If time permits I will introduce a moduli space of semiorthogonal decompositions, and explain how this indecomposability result follows from its geometric properties. This is joint work in progress with Shinnosuke Okawa and Andrea Ricolfi.
Projections and jumps between Grassmannians and Fano varieties of CY type
by Marcello Bernardara

Abstract: Using geometrical correspondences, we relate the Hodge structure and the derived categories of subvarieties of different Grassmannians. This leads to show that the CY subHodge structure of hyperplane sections of $$G(3,n)$$ is isomorphic to the CY subHodge structure of other examples. Similar results hold conjecturally for CY-subcategories. I will give explicit examples in the K3 case arising from $$G(3,10)$$, and explain also how these correspondences allow to construct a crepant categorical resolution for the Coble cubic. All these results are joint with Fatighenti and Manivel.
An easy characterization of highest weight categories
by Agnieszka Bodzenta

Abstract: I will recall the definition of a highest weight category and prove that an abelian category A is highest weight if and only if its derived category has a full exceptional collection of objects of A whose left dual collection also consists of objects of A. I will use this criterion to attach a highest weight category to a birational morphism of smooth surfaces. This is joint work with A. Bondal.
Formality of $$\mathbb{P}$$-objects
by Andreas Hochenegger

Abstract: An Calabi-Yau-object in a $$k$$-linear triangulated category is called a $$\mathbb{P}$$-object, if its derived endomorphism ring is isomorphic to $$k[t]/t^n$$. They were first studied by Daniel Huybrechts and Richard Thomas as generalisations of spherical objects. Similar to the spherical case, $$\mathbb{P}$$-objects induce autoequivalences which are called $$\mathbb{P}$$-twists. Ed Segal showed how an arbitrary autoequivalence can be written as a spherical functor. For a $$\mathbb{P}$$-twist, he needs the assumption that the endomorphism ring of the $$\mathbb{P}$$-object is formal. In this talk, I will introduce the concept of formality and present a proof of the formality of $$\mathbb{P}$$-objects. This is based on a joint work with Andreas Krug.
Adjunction in 2-categories
by Dmitry Kaledin

Abstract: As the theory of spherical functors shows, it pays to take adjunction seriously, and even studying it in a general 2-categorical context might be a worthwhile exercise. Unfortunately, the study of 2-categories is traditionally mired in all sorts of multishaped multidimensional commutative diagrams, exotic notation, obscure rigidification theorems and general unpleasantness. I will try to show that with the right packaging, the theory is actually quite concise and workable, up to and including adjunctions, and meaningful new results are not beyond reach.
Rationality of Fano 3-folds over non-closed fields
by Alexander Kuznetsov

Abstract: In the talk I will discuss rationality criteria for Fano 3-folds of geometric Picard number 1 over a non-closed field $$k$$ of characteristic $$0$$. Among these there are 8 types of geometrically rational varieties. We prove that in one of these cases any variety of this type is $$k$$-rational, in four cases the criterion of rationality is the existence of a $$k$$-rational point, and in the last three cases the criterion is the existence of a $$k$$-rational point and a $$k$$-rational curve of genus 0 and degree 1, 2, and 3 respectively. The last result is based on recent results of Benoist-Wittenberg. This is a joint work with Yuri Prokhorov.
Hochschild cohomology of semi-co-categories
by Wendy Lowen

Abstract: We introduce a new tool to establish and compare higher structure on Hochschild-type complexes, inspired by Keller's arrow category which he famously used to obtain derived invariance for dg categories. As an application, we compare the Hochschild complex of an algebra with the co-Hochschild complex of its Bar resolution. This is joint work with Michel Van den Bergh.
The Lie algebra acting on the cohomology of a projective variety (revisited)
by Valery Lunts

Abstract: I will recall the construction and properties of the Lie algebra defined jointly with E. Looijenga and independently by M. Verbitsky. Then I will review its remarkable connection with the group of autoequivalences of the derived category, and formulate some conjectures.
On the geometry of contractions of the moduli space of sheaves on a K3 surface
by Diletta Martinelli

Abstract: I will describe how recent advances have made possible to study the birational geometry of hyperkaehler varieties of K3-type using the machinery of wall-crossing and stability conditions on derived categories as developed by Tom Bridgeland. In particular Bayer and Macrì relate birational transformations of the moduli space $$M$$ of sheaves on a K3 surface $$X$$ to wall-crossing in the space of Bridgeland stability conditions $$\mathrm{Stab}(X)$$. I will explain how it is possible to refine their analysis to give a precise description of the geometry of the exceptional locus of any birational contraction of $$M$$.
K3 categories in Fano varieties of K3 type
by Giovanni Mongardi

Abstract: Fano varieties of K3 type are of interest for their deep links with hyperkahler geometry and K3 categories. In this talk, I will mainly illustrate the construction of some new examples of such varieties and the hyperkahler and categorical information they carry. This is a joint work with Enrico Fatighenti.
The integral Hodge conjecture for CY2 categories
by Alex Perry

Abstract: I will discuss a proof of a version of the integral Hodge conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface. This has applications to the integral Hodge conjecture for varieties whose Kuznetsov component is such a category.
Stability conditions on Gushel-Mukai fourfolds
by Laura Pertusi

Abstract: An ordinary Gushel-Mukai fourfold $$X$$ is a smooth quadric section of a linear section of the Grassmannian $$G(2,5)$$. Kuznetsov and Perry proved that the bounded derived category of $$X$$ admits a semiorthogonal decomposition whose non-trivial component is a subcategory of K3 type. In this talk I will report on a joint work in progress with Alex Perry and Xiaolei Zhao, where we construct Bridgeland stability conditions on the K3 subcategory of $$X$$. Then I will explain some applications concerning the existence of a homological associated K3 surface and hyperkaehler geometry.
Kawamata type semiorthogonal decompositions for singular varieties
by Evgeny Shinder

Abstract: After introducing Kawamata type decompositions and recollecting known examples for curves, surfaces and threefolds due to Burban, Kawamata, Kuznetsov and Karmazyn-Kuznetsov and myself, I will explain a general approach to obstructions for such decompositions based on negative K-theory, Orlov's singularity category and its Grothendieck group recently investigated by Pavic and myself. Applying these obstructions to threefolds, it turns out that most types of nodal Fano threefolds do not admit Kawamata type decompositions. The talk is based on joint work in progress with M. Kalck and N. Pavic.
Residual categories of Grassmannians
by Maxim Smirnov

Abstract: I will define residual categories of Lefschetz decompositions and discuss a conjectural relation between the structure of quantum cohomology and residual categories. I will illustrate this relationship in the case of some isotropic Grassmannians. This is a joint work with Alexander Kuznetsov.
A class of perverse schobers in geometric invariant theory
by Špela Špenko

Abstract: Perverse schobers are (conjectural) categorifications of perverse sheaves. We construct perverse schobers on a partial compactification of the stringy Kähler moduli space associated to the GIT quotient stack for an action of a reductive group on a finite-dimensional (quasi-symmetric) representation. This is a joint work with Michel Van den Bergh.
Categorical Torelli Theorems
by Paolo Stellari

Abstract: We investigate a refined Derived Torelli Theorem for Enriques surfaces. Namely, we prove that two (generic) Enriques surfaces are isomorphic if and only if their Kuznetsov components are Fourier-Mukai equivalent. We investigate the similarities with analogous results for cubic fourfolds and threefolds and we show the applications of our techniques to a conjecture by Ingalls and Kuznetsov about the derived categories of Artin-Mumford quartic double solids. This is joint work in progress with Li, Nuer and Zhao.
Tensor-triangular geometry of relative stable categories
by Greg Stevenson

Abstract: I'll discuss joint work with Paul Balmer which proposes a general method to construct new triangulated categories as additive quotients of a given one, enhancing results of Beligiannis, particularly in the tensor-triangular setting. We prove a birationality result showing that the resulting quotient tt-category coincides with the given one on some open piece of the spectrum.
Noncommutative Weil conjectures
by Gonçalo Tabuada

Abstract: The Weil conjectures (proved by Deligne in the 70's) played a key role in the development of modern algebraic geometry. In this talk, making use of some recent topological "technology", I will extended the Weil conjectures from the realm of algebraic geometry to the broad noncommutative setting of differential graded categories. Moreover, I will prove the noncommutative Weil conjectures in some interesting cases.
The Frobenius morphism in invariant theory
by Michel Van den Bergh

Abstract: We will report on joint work with Špela Špenko and Theo Raedschelders. We will present a number of general conjectures concerning the Frobenius push forward of rings of invariants for reductive groups. We will show that these conjectures are true for the invariants of the $$\mathrm{SL}_2$$-action on a number of copies of the standard representation.
$$K(\pi,1)$$ via spherical functors
by Michael Wemyss

Abstract: Usually in algebraic geometry settings, such as for a flopping contraction $$X \to \mathrm{Spec} \ R$$, we are interested in establishing contractibility of the stability manifold. Actually, in practice this is not strictly true: we are usually interested in proving contractibility of certain components of the stability manifold of certain subcategories of $$\mathrm{D^b}(\mathrm{coh} \ X)$$. Depending on which components, and on which subcategories, the difficulty of the problem varies. I will explain how to approach the contractibility in one such case, by "mirroring" the stability manifold from the image of a spherical functor, to the source category of the spherical functor. The source category is conjecturally giving the classification of flops, so it should behave well. It does! Contractibility is much easier there. There are three main corollaries: $$K(\pi,1)$$ for all intersection arrangements in ADE root systems (which includes the Coxeter groups $$I_n$$ with $$n=3,4,5,6,8$$), plus faithfulness of group actions in various settings (the first avoiding normal forms), plus contractibility of stability manifolds in some 3-CY settings. This is joint work with Jenny August.