Derived categories and Kummer varieties

We prove that if two abelian varieties have equivalent derived categories then the derived categories of the smooth stacks associated to the corresponding Kummer varieties are equivalent as well. The second main result establishes necessary and sufficient conditions for the existence of equivalences between the twisted derived categories of two Kummer surfaces in terms of Hodge isometries between the generalized transcendental lattices of the corresponding abelian surfaces.Comment: 14 pages. Minor changes (theorem numbering slightly changed). Final version to appear in Math. Zei

A finite group acting on the moduli space of K3 surfaces

We consider the natural action of a finite group on the moduli space of polarized K3 surfaces which induces a duality defined by Mukai for surfaces of this type. We show that the group permutes polarized Fourier-Mukai partners of polarized K3 surfaces and we study the divisors in the fixed loci of the elements of this finite group.Comment: 10 pages, major revisions, final version to appear in Trans. Amer. Math. So

Some remarks about the FM-partners of K3 surfaces with Picard numbers 1 and 2

In this paper we prove some results about K3 surfaces with Picard number 1 and 2. In particular, we give a new simple proof of a theorem due to Oguiso which shows that, given an integer $N$, there is a K3 surface with Picard number 2 and at least $N$ non-isomorphic FM-partners. We describe also the Mukai vectors of the moduli spaces associated to the Fourier-Mukai partners of K3 surfaces with Picard number 1.Comment: LaTeX, 10 page

Uniqueness of dg enhancements for the derived category of a Grothendieck category

We prove that the derived category of a Grothendieck abelian category has a unique dg enhancement. Under some additional assumptions, we show that the same result holds true for its subcategory of compact objects. As a consequence, we deduce that the unbounded derived category of quasi-coherent sheaves on an algebraic stack and the category of perfect complexes on a noetherian concentrated algebraic stack with quasi-finite affine diagonal and enough perfect coherent sheaves have a unique dg enhancement. In particular, the category of perfect complexes on a noetherian scheme with enough locally free sheaves has a unique dg enhancement.Comment: 32 pages. This version fills a gap in the proof of the previous Theorem C which now appears with slightly stronger hypotheses. The main results (Theorems A and B and their geometric applications) are unchange

Automorphisms and autoequivalences of generic analytic K3 surfaces

This is a systematic exposition of recent results which completely describe the group of automorphisms and the group of autoequivalences of generic analytic K3 surfaces. These groups, hard to determine in the algebraic case, admit a good description for generic analytic K3 surfaces, and are in fact seen to be closely interrelated.Comment: 34 pages. Minor corrections. Final version to appear in J. Geom. Phy

Twisted Fourier-Mukai functors

Due to a theorem by Orlov every exact fully faithful functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of Fourier-Mukai type. We extend this result to the case of bounded derived categories of twisted coherent sheaves and at the same time we weaken the hypotheses on the functor. As an application we get a complete description of the exact functors between the abelian categories of twisted coherent sheaves on smooth projective varieties.Comment: 16 pages. Minor changes. Final version to appear in Adv. Mat

Non-uniqueness of Fourier-Mukai kernels

We prove that the kernels of Fourier-Mukai functors are not unique in general. On the other hand we show that the cohomology sheaves of those kernels are unique. We also discuss several properties of the functor sending an object in the derived category of the product of two smooth projective schemes to the corresponding Fourier-Mukai functor.Comment: 12 pages. Minor changes. Final version to appear in Math. Zei

Equivalences of twisted K3 surfaces

We prove that two derived equivalent twisted K3 surfaces have isomorphic periods. The converse is shown for K3 surfaces with large Picard number. It is also shown that all possible twisted derived equivalences between arbitrary twisted K3 surfaces form a subgroup of the group of all orthogonal transformations of the cohomology of a K3 surface. The passage from twisted derived equivalences to an action on the cohomology is made possible by twisted Chern characters that will be introduced for arbitrary smooth projective varieties.Comment: Final version. 35 pages. to appear in Math. An

Fano varieties of cubic fourfolds containing a plane

We prove that the Fano variety of lines of a generic cubic fourfold containing a plane is isomorphic to a moduli space of twisted stable complexes on a K3 surface. On the other hand, we show that the Fano varieties are always birational to moduli spaces of twisted stable coherent sheaves on a K3 surface. The moduli spaces of complexes and of sheaves are related by wall-crossing in the derived category of twisted sheaves on the corresponding K3 surface.Comment: 26 pages. Many changes in the exposition. Final version to appear in Math. An