• Algebraic Geometry
• Workshop on derived categories

• Schedule
• Participants

This is a joint work in progress with A. Auel. Let S be a geometrically rational del Pezzo surface over a field k. In this talk, I will show how the k-rationality of S is equivalent to the existence of some semiorthogonal decompositions of its derived category. In particular, the surface S is k-rational if and only if it is categorically representable in dimension 0. Moreover, if the degree of S is at least 5, semiorthogonal decompositions allow to define a pair of algebras which give a birational invariant for S. As a corollary, one can show that a minimal non-rational del Pezzo of degree 6 is semirigid. One of the main tools in the proof is the classification of 3-block exceptional collections on complex del Pezzo surfaces given by Karpov and Nogin.

In this talk I will introduce the definition of categorical representability for a projective variety. I will discuss its relevance with respect to rationality problems for varieties of dimension up to 4 by showcasing several examples developed in the recent years with A. Auel, M. Bernardara and T. Varilly-Alvarado.

Nodal rings are non-commutative analogues of the -singularity . I shall show that to any (non-commutative) rational projective curve one can canonically attach a finite dimensional algebra satisfying the following properties:

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•there exists a fully faithful exact functor

• and have the same (derived) representation type.

In these terms we get a geometric realization of a broad class of derived tame algebras which include degenerate tubular algebras, certain classes of gentle, skew-gentle and supercanonical algebras as well as some new examples of derived tame algebras.

In a similar way I shall show that any non-commutative rational projective curve can be categorically resolved by a finite dimensional algebra of finite global dimension.

This is a joint work in progress with Yuriy Drozd and Volodymyr Gavran.

Motivated by joint work with Perling on rational surfaces and results of Böhning, von Bothmer and Sosna on surfaces of general type, we are interested in a deeper and more detailed understanding of the derived category of an abelian category of global dimension two. In fact, we can describe the objects by passing to a certain category of matrices. This category is hereditary, and, in a certain sense, free. Under some additional condition, it is even an additive subcategory of an abelian category. We explain the result and mention certain surprising applications. In fact we can give explanations for several well-known results in a different way. Also note that the derived category of a hereditary abelian category is well-known by a classical result of Happel. Our result looks like the next step, however we show that many cases can be reduced to global dimension two. We finally mention several examples, where we can even explicitly describe all proper complexes of the derived category.

Equivalences of derived categories of coherent sheaves on smooth varieties having non-vanishing first Betti number have been recently studied by M. Popa and C. Schnell. An important problem in this subject is a conjecture of Popa predicting the derived invariance of all the non-vanishing loci associated to the canonical bundle introduced by M. Green and R. Lazarsfeld. One of the main results I will present in the talk is that Popa's conjecture is equivalent to the problem concerning the invariance of the Hodge numbers for derived equivalent smooth projective varieties. As an application I will verify Popa's conjecture for varieties of dimension up to three and I will present a few results concerning the behavior of fibrations onto smooth curves of genus at least 2 under equivalences of derived categories.
This is a joint work with Mihnea Popa.

This talk will elaborate on the role hyperholomorphic sheaves play in generalized deformations of K3 surfaces, described in the talk of Sukhendu Mehrotra.

The Hilbert scheme of points of a K3 surface X admits a 21-dimensional space of deformations, while the moduli space of K3 surfaces is 20-dimensional. The goal of this talk is to provide an interpretation of this extra modulus of the deformation space of the Hilbert scheme X^[n] in terms of deformations of the derived category D(Coh(X)). This is joint work with Eyal Markman (UMass).

In this talk I will present some results obtained in a joint work with Matei Toma about moduli spaces of stable sheaves on non-algebraic K3 surfaces: we show that if the Mukai vector of the sheaves is of the form with and prime to each other, and if the polarization is generic in the Kahler cone, then the moduli space of -slope stable sheaves with Mukai vector is an irreducible hyperkahler manifold which is deformation equivalent to a Hilbert scheme of points on a projective K3. Moreover, we calculate the Beauville form, and show that the moduli space is projective if and only if the base surface is projective. The proof uses moduli spaces of twisted sheaves and Hitchin's characterisation of twistor families of hyperkahler manifolds. As intermediate result we show that all these moduli spaces are Kahler, and we describe their twistor family (even in the projective context).

We analyze exceptional sequences on rational surfaces from a combinatorial point of view. We show that to each such sequences there is naturally associated a toric surface with T-singularities. We also show that by mutation, any such sequences can be transformed into a sequence consisting only of objects of rank one.

We establish generators for the group of autoequivalences of Hilbert schemes of the projective plane. The standard equivalences of neither of the two canonical hearts suffice, but together they do. The method also works for some other surfaces. (This is work in progress, jointly with Andreas Krug.)

A quasi-phantom category is an admissible category in the bounded derived category of a smooth projective variety having trivial Hochschild homology and finite Grothendieck group. If, in addition, the Grothendieck group vanishes, then we call such a category a phantom.
In this talk I will review several recent constructions of (quasi-)phantom categories and discuss some connections to the rationality question of cubic fourfolds.

Fourier-Mukai functors play a distinct role in algebraic geometry. Nevertheless a basic question is still open: are all exact functors between the bounded derived categories of smooth projective varieties of Fourier-Mukai type? We discuss the recent advances in the subject and study the same question in the context of dg categories where the problem has been settled by B. Toën. In this talk we propose a simpler approach not based on the notion of model category. This is a joint work with A. Canonaco.

A description of a distinguished connected component of the space of stability conditions of certain CY3 triangulated categories associated to moduli spaces of meromorphic flat SL(2,C)-connections on Riemann surfaces was recently given by Bridgeland and Smith. The categories can be constructed as the bounded derived categories of the Ginzburg dg algebras associated to the dimer models given by the Stokes curves of a flat connection.  We will give an explicit computation of these spaces of stability conditions in the case of the moduli spaces of lowest possible quaternionic dimension one, which are complements of an anti-canonical divisor in a rational surface.

Hilbert schemes of points on projective plane admit deformations, which were constructed by Nevins and Stafford. I will explain this construction, and report on my recent joint work with Li, in which we study the birational models of these deformations using wall crossing in Bridgeland stability space.