Showing papers by "Daniel Huybrechts published in 2008"
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TL;DR: In this article, a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family is given, which is a product of Atiyah and Kodaira--Spencer classes.
Abstract: We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that it is a product of Atiyah and Kodaira--Spencer classes. This allows us to obtain deformation-invariant virtual cycles on moduli spaces of objects of the derived category on threefolds.
100 citations
TL;DR: In this paper, the authors studied stability conditions on K3 categories and proved the topology of the stability manifold and the autoequivalences group for generic twisted projective K3, abelian surfaces, and K3 surfaces with trivial Picard group.
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.
81 citations
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TL;DR: In this article, the general fiber of a formal deformation over the formal disk of a projective variety from the view point of abelian and derived categories is studied, and it is shown that the derived category is a linear triangulated category with a Serre functor given by the square of the shift functor.
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.
4 citations
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TL;DR: In this paper, it was shown that the decomposition of a Ricci-flat manifold in its irreducible factors is reflected by the derived category of coherent sheaves, and a smooth projective variety that is derived equivalent to an abelian variety is of the same type.
Abstract: We present results indicating that the decomposition of a Ricci-flat manifold in its irreducible factors is reflected by the derived category of coherent sheaves. More precisely, we prove that a smooth projective variety that is derived equivalent to an abelian variety resp. an irreducible symplectic variety is of the same type.
The paper also contains a proof of a conjecure of Caldararu for manifolds with trivial canonical bundle saying that the modified HKR isomorphism for Hochschild homology is compatible with the module structure.
3 citations
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TL;DR: In this article, it was shown that the finitely generated subring R(X) of the Chow ring introduced by Beauville and Voisin is preserved under derived equivalences.
Abstract: We show that for a K3 surface X the finitely generated subring R(X) of the Chow ring introduced by Beauville and Voisin is preserved under derived equivalences This is proved by analyzing Chern characters of spherical bundles As for a K3 surface X defined over a number field all spherical bundles on the associated complex K3 surface are defined over $\bar\QQ$, this is compatible with the Bloch-Beilinson conjecture Besides the work of Beauville and Voisin, Lazarfeld's result on Brill-Noether theory for curves in K3 surfaces and the deformation theory developed with Macri and Stellari in arXiv:07101645 are central for the discussion
1 citations