Paolo Stellari - Research

Papers and preprints

This page contains the latest versions of my papers and preprints. Most of them are also present on ArXiv and Mathscinet (in non-updated versions).

Click on the title to see the abstract.

A. Canonaco, P. Stellari, Fourier-Mukai functors: a survey, arXiv:1109.3083 (pdf).
Abstract. This paper surveys some recent results about Fourier-Mukai functors. In particular, given an exact functor between the bounded derived categories of coherent sheaves on two smooth projective varieties, we deal with the question whether this functor is of Fourier-Mukai type. Several related questions are answered and many open problems are stated.

A. Canonaco, D. Orlov, P. Stellari, Does full imply faithful?, to appear in: J. Noncommut. Geom., arXiv:1101.5931 (pdf).
Abstract. We study full exact functors between triangulated categories. With some hypotheses on the source category we prove that it admits an orthogonal decomposition into two pieces such that the functor restricted to one of them is zero while the restriction to the other is faithful. In particular, if the source category is either the category of perfect complexes or the bounded derived category of coherent sheaves on a noetherian scheme supported on a closed connected subscheme, then any non-trivial exact full functor is faithful as well. Finally we show that removing the noetherian hypothesis this result is not true.

A. Canonaco, P. Stellari, Fourier-Mukai functors in the supported case, arXiv:1010.0798 (pdf).
Abstract. We prove that exact functors between the categories of perfect complexes supported on projective schemes are of Fourier-Mukai type if the functor satisfies a condition weaker than being fully faithful. We also get generalizations of the results in the literature in the non-supported case. Some applications are discussed and, along the way, we prove that the category of perfect supported complexes has a strongly unique enhancement.

A. Canonaco, P. Stellari, Non-uniqueness of Fourier-Mukai kernels, to appear in: Math. Z., arXiv:1009.5577 (pdf).
Abstract. 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.

E. Macrì, P. Stellari, Fano varieties of cubic fourfolds containing a plane, to appear in: Math. Ann., arXiv:0909.2725 (pdf).
Abstract. 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.

M. Bernardara, E. Macrì, S. Mehrotra, P. Stellari, A categorical invariant for cubic threefolds, Adv. Math. 229 (2012), 770-803 (pdf).
Abstract. We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class.

D. Huybrechts, E. Macrì, P. Stellari, Formal deformations and their categorical fibre, Comment. Math. Helv. 86 (2011), 41-71 (pdf).
Abstract. We study the general fibre of a formal deformation over the formal disk of a projective variety from the view point of abelian and derived categories. The abelian category of coherent sheaves of the general fibre is constructed directly from the formal deformation and is shown to be linear over the field of Laurent series. The various candidates for the derived category of the general fibre are compared. If the variety is a surface with trivial canonical bundle, we show that the derived category of the general fibre is again a linear triangulated category with a Serre functor given by the square of the shift functor.

D. Huybrechts, E. Macrì, P. Stellari, Derived equivalences of K3 surfaces and orientation, Duke Math. J. 149 (2009), 461-507 (pdf).
Abstract. Every Fourier-Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing. We prove that this Hodge isometry preserves the natural orientation of the four positive directions. This leads to a complete description of the action of the group of all autoequivalences on cohomology very much like the classical Torelli theorem for K3 surfaces determining all Hodge isometries that are induced by automorphisms.

E. Macrì, P. Stellari, Infinitesimal derived Torelli theorem for K3 surfaces, (Appendix by S. Mehrotra), Int. Math. Res. Not. IMRN 2009, 3190-3220 (pdf).
Abstract. We prove that the first order deformations of two smooth projective K3 surfaces are derived equivalent under a Fourier-Mukai transform if and only if there exists a special isometry of the total cohomology groups of the surfaces which preserves the Mukai pairing, an infinitesimal weight-2 decomposition and the orientation of a positive 4-dimensional space. This generalizes the derived version of the Torelli Theorem. Along the way we show the compatibility of the actions on Hochschild homology and singular cohomology of any Fourier-Mukai functor.

E. Macrì, S. Mehrotra, P. Stellari, Inducing stability conditions, J. Algebraic Geom. 18 (2009), 605-649 (pdf).
Abstract. We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a s mooth projective variety we prove that the subset of invariant stability conditions embeds as a closed submanifold into the stability manifold of the equivariant derived category. As an application we examine stability conditions on Kummer and Enriques surfaces and we improve the derived version of the Torelli Theorem for the latter surfaces already present in the litterature. We also study the relationship between stability conditions on projective spaces and those on their canonical bundles.

E. Macrì, M. Nieper-Wisskirchen, P. Stellari, The module structure of Hochschild homology in some examples, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 863-866 (pdf).
Abstract. In this note we give a simple proof of a conjecture by A. Căldăraru stating the compatibility between the modified Hochschild-Kostant-Rosenberg isomorphism and the action of Hochschild cohomology on Hochschild homology in the case of Calabi-Yau manifolds and smooth projective curves.

P. Stellari, A finite group acting on the moduli space of K3 surfaces, Trans. Amer. Math. Soc. 360 (2008), 6631-6642 (pdf).
Abstract. 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.

E. Macrì, P. Stellari, Automorphisms and autoequivalences of generic analytic K3 surfaces, J. Geom. Phys. 58 (2008), 133-164 (pdf).
Abstract. 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.

D. Huybrechts, E. Macrì, P. Stellari, Stability conditions for generic K3 categories, Compositio Math. 144 (2008), 134-162 (pdf).
Abstract. A K3 category is by definition a Calabi-Yau category of dimension two. Geometrically K3 categories occur as bounded derived categories of (twisted) coherent sheaves on K3 or abelian surfaces. A K3 category is generic if there are no spherical objects (or just one up to shift). We study stability conditions on K3 categories as introduced by Bridgeland and prove his conjecture about the topology of the stability manifold and the autoequivalences group for generic twisted projective K3, abelian surfaces, and K3 surfaces with trivial Picard group.

A. Canonaco, P. Stellari, Twisted Fourier-Mukai functors, Adv. Math. 212 (2007), 484-503 (pdf).
Abstract. 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.

P. Stellari, Derived categories and Kummer varieties, Math. Z. 256 (2007), 425-441 (pdf).
Abstract. 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.

D. Huybrechts, P. Stellari, Proof of Căldăraru's conjecture. An appendix to a paper by K. Yoshioka, The 13th MSJ Inter. Research Inst. - Moduli Spaces and Arithmetic Geometry, Adv. Stud. Pure Math. 45 (2006), 31-42 (pdf).
Abstract. In this short note we show how to combine Yoshioka's recent results on moduli spaces of twisted sheaves on K3 surfaces to prove Căldăraru's conjecture on the equivalence of twisted derived categories of projective K3 surfaces.

D. Huybrechts, P. Stellari, Equivalences of twisted K3 surfaces, Math. Ann. 332 (2005), 901-936 (pdf).
Abstract. 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.

P. Stellari, Some remarks about the FM-partners of K3 surfaces with Picard numbers 1 and 2, Geom. Dedicata 108 (2004), 1-13 (pdf).
Abstract. 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.

P. Stellari, Singular cubic fourfolds containing a plane, Le Matematiche Vol. LVIII (2003), 277-305 (pdf).
Abstract. In this paper we consider cubic 4-folds containing a plane whose discriminant curve is a reduced nodal plane sextic. In particular, we describe the singular points of such cubic 4-folds and we give an estimate of the rank of the free abelian group generated by the equivalence classes of the algebraic cycles of codimension 2. Moreover, we describe some conditions on the geometry of the plane sextics so that all the associated cubic 4-folds are singular and we construct a family of smooth rational cubic 4-folds whose discriminant curve is reduced but reducible.