Derived Categories in Algebraic Geometry: mini-courses, talks and abstracts

September 5-9, 2011,
Laboratory of Algebraic Geometry,
Steklov Math Institute,

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Henning Krause (Bielefeld University)
Stratifying triangulated categories

For a compactly generated triangulated category, there is a notion of stratification based on the action of a graded commutative noetherian ring. I will explain this concept and its applications. Then I will present examples where such a stratification has been established. This is a report on a joint project with Dave Benson and Srikanth Iyengar.

Alexander Polishchuk (University of Oregon)
Fan-Jarvis-Ruan Theory

I will describe recent joint work with Arkady Vaintrob. Fan-Jarvis-Ruan theory is an analog of Gromov-Witten theory in which a target space is replaced by a quasihomogeneous isolated hypersurface singularity. In the case of simple singularities of type A it corresponds to the intersection theory on the moduli space of higher spin curves, and constitutes a framework for the famous Witten conjectures, proved by Kontsevich for A_1-singularity and by Faber-Shadrin- Zvonkine in general. In my lectures I will explain the algebro-geometric construction of the relevant cohomological field theory based on the theory of matrix factorizations. The crucial construction of an analog of the virtual fundamental class involves derived categories of matrix factorizations in a global setting.


Alexei Bondal (Steklov Institute)
Orthogonal decompositions of sl(n) and mutually unbiased bases

Igor Burban (University of Bonn)
Cohen-Macaulay modules over non-isolated surface singularities

This talk is based on my joint work with Yuriy Drozd, arXiv:1002.3042.

I am going to explain the classification of indecomposable Cohen-Macaulay modules over a certain class of non-isolated Gorenstein surface singularities called degenerate cusps. The core of our approach is a categorical construction, allowing to reduce this classification problem to a certain matrix problem of tame representation type.

I am going to illustrate our method on the case of the rings k[[x,y,z]]/(x^3 + y^2 - xyz), k[[x,y,z]]/(xyz) and k[[x,y,u,v]]/(xy, uv). In the case of degenerate cusps, which are hypersurface singularities, our technique leads to a description of all indecomposable matrix factorizations of the underlying polynomials.

Alberto Canonaco (Università di Pavia)
Non-uniqueness of Fourier-Mukai kernels

We will discuss some properties of the functor sending an object in the bounded derived category of the product of two smooth projective varieties to the Fourier-Mukai functor with that object as kernel. In particular we will show that this functor is not always essentially injective, namely that there exist non-isomorphic kernels defining isomorphic Fourier-Mukai functors. On the other hand, the cohomology sheaves of a kernel are always uniquely determined, up to isomorphism, by the functor. This is a joint work with P. Stellari.

Tobias Dyckerhoff (Yale University)
Higher Segal spaces

I will describe joint work in progress with Mikhail Kapranov.

Segal spaces were introduced by C. Rezk as models for higher categories whose composition law is weakly associative up to coherent homotopy. In this talk, we will introduce the notion of a 2-Segal space which is a simplicial space satisfying a weaker locality condition than a (1-)Segal space. We will discuss examples of 2-Segal spaces, such as Waldhausen spaces, explain their relevance for Hall algebra constructions, and outline their basic theory within the context of model categories.

Alexei Elagin (IITP, Moscow)
Equivariant derived categories and descent theory for derived categories

I will talk about derived categories of equivariant sheaves on algebraic varieties. A series of examples of full exceptional collections in such categories will be presented, as well as general method to construct them. The construction starts from an exceptional collection in ordinary derived category and produces an exceptional collection in equivariant derived category by descent. In this context, I will give a simple but useful criterion for the derived category of a variety to be recovered from the derived category of its covering by means of descent.

David Favero (University of Vienna)
Generation time and algebraic cycles

Generation time was introduced by A. Bondal and M. Van den Bergh in order to demonstrate saturatedness of the derived category of a smooth proper variety. The notion was developed further by R. Rouquier and D. Orlov where they study the minimal generation time and the set of all generation times as a categorical invariant. I will summarize results in a recent preprint with M. Ballard and L. Katzarkov connecting generation time to the existence of algebraic cycles on a smooth variety.

Daniel Huybrechts (Universitaet Bonn)
Chow groups of K3 surfaces

I review some of the open problems concerning the Chow groups of K3 surfaces over the complex numbers and over number fields. In particular, I shall discuss the case of generic Picard group in the various settings (eg for symplectic automorphisms)

Timothy Logvinenko (University of Warwick)
Relative spherical objects

Seidel and Thomas introduced some years ago a notion of a spherical object in the derived category D(X) of a smooth projective variety X. Such objects induce, in a simple way, auto-equivalences of D(X) called 'spherical twists'. In a sense, they are mirror-symmetric analogues of Lagrangian spheres on a symplectic manifold and the induced auto-equivalences mirror the Dehn twists associated with the latter. We generalise this notion to the relative context by explaining what does it mean for an object of D(Z x X) to be _spherical over Z_ for any two separated schemes Z and X of finite type. This is a joint work with Rina Anno.

Wendy Lowen (Universiteit Antwerpen)
On compact generation of some deformed surfaces

We discuss applications of Rouquier's cocovering theorem, which allows to prove compact generation of triangulated categories based upon certain coverings by localizations, to the setting of Grothendieck categories. We obtain some positive results for Grothendieck categories arising as non-commutative deformations of certain surfaces.

Emanuele Macri (University of Bonn)
Bogomolov-type inequalities in higher dimension

In this seminar (based on joint work in progress with A. Bayer, Y. Toda, and A. Bertram), we will present a conjectural approach to the construction of Bridgeland stability conditions on the derived category of a higher dimensional variety. The main ingredient is a generalization to complexes of the classical Bogomolov inequality for sheaves.

We will also discuss an application of this inequality to the Fujita Conjecture for threefolds.

Amnon Neeman (Australian National University)
The spectrum of a triangulated category acted on by another

There has been a string of recent results relating tensor triangulated categories with algebro-geometric objects, known as their "spectra". I will report on the recent PhD thesis of my student Greg Stevenson, who developed a variant of this theory when one triangulated category acts on another. His main application is to Orlov's derived category of singularities.

Leonid Positselsky (HSE)
Matrix factorizations and exotic derived categories

The triangulated category of conventional free or projective matrix factorizations over an affine scheme can be simply defined as their homotopy category, but the constructions of the derived categories of the second kind are needed in order to work either with locally free matrix factorizations over a nonaffine scheme, or with coherent (analogues of) matrix factorizations. I will discuss the definitions of the coderived and absolute derived categories of various classes of matrix factorizations, and their interrelations. Various complements to and strengthenings of Orlov's recent theorem connecting matrix factorizations over a singular nonaffine scheme with the triangulated category of singularities of the zero locus of the superpotential will be given.

Paolo Stellari (Universita' degli Studi di Milano)
Fourier-Mukai functors in the supported setting

Fourier-Mukai functors play a distinct role in algebraic geometry. Nevertheless two questions remained open: are all exact functors between the bounded derived categories of smooth projective varieties of Fourier-Mukai type? Is the Fourier-Mukai kernel unique? We will answer positively to these questions under some assumptions on the exact functor. This extends previous results by Lunts, Orlov and Ballard. Along the way, we will show that, in geometric contexts, full functors are faithful as well. This is a joint work in collaboration with A. Canonaco and, partly, with D. Orlov.

Yukinobu Toda (University of Tokyo, IPMU)
Multiple cover formula of generalized DT invariants

The generalized DT invariants counting one dimensional semistable sheaves on Calabi-Yau 3-folds are conjectured to satisfy a certain multiple cover formula. In this talk, I explain an approach toward the multiple cover conjecture using Jacobian localizations and the notion of parabolic stable pairs.

Vadik Vologodsky (University of Oregon)
On the Hodge realization of Voevodsky's motives

I will explain how a very general construction from Homological Algebra yields a functor from the category of Voevodsky's motives over the field of complex numbers to a certain derived category of mixed Hodge structures with the weight filtration defined integrally.

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Laboratory of Algebraic Geometry and its Applications