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Daniel Huybrechts

Publications and source records attributed to Daniel Huybrechts.

52 records · Page 3Linked to original sources
arXiv2008

Remarks on derived equivalences of Ricci-flat manifolds

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.

math.AG↗
arXiv2007

Stability conditions for generic K3 categories

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.

math.AG↗
arXiv2006

Derived and abelian equivalence of K3 surfaces

Tom Bridgeland constructed explicit stability conditions on K3 surfaces. This paper attempts to shed more light on these particular examples, especially on the hearts of the underlying t-structures. We prove that two K3 surfaces X and X' are derived equivalent if and only if there exist complexified polarizations B+iw and B'+iw' such that the associated abelian categories A(B+iw) and A(B'+iw') are equivalent. We study in detail the minimal objects of A(B+iw) and investigate stability under Fourier-Mukai transform.

math.AG↗
arXiv2006

Proof of Caldararu's conjecture. An appendix to a paper by Yoshioka

We show that any orientation preserving Hodge isometry between the Hodge structures of two K3 surfaces X and X' twisted by Brauer classes $α$ resp. $α'$ can be lifted to a Fourier-Mukai equivalence between the derived categories of $α$ resp. $α'$-twisted sheaves.

math.AG↗
arXiv2005

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.

math.AG↗
arXiv2004

Generalized Calabi-Yau structures, K3 surfaces, and B-fields

Generalized Calabi-Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are both special cases of this general notion. The moduli space of generalized Calabi-Yau structures admits a canonical symplectic form with respect to which the moduli space of symplectic structures is Lagrangian. The standard theory of K3 surfaces implies surjectivity of the period map and a weak form of the Global Torelli theorem.

math.AG↗
arXiv2003

Moduli spaces of hyperkaehler manifolds and mirror symmetry

These are notes of my lectures given at the school on intersection theory and moduli at the ICTP, Trieste. We construct moduli spaces of K3 surfaces and higherdimensional hyperkaehler manifolds, including moduli spaces of (2,2)-conformal field theories associated to hyperkaehler manifolds, and describe their period domains. Recent results on these manifolds are discussed from the moduli space point of view. The second goal is to present a detailed account of mirror symmetry for K3 surfaces. Arguments due to Aspinwall and Morrison allow one to explain various versions of mirror symmetry (e.g. for lattice polarized or elliptic K3 surfaces) by the same general principle.

math.AG↗
arXiv2003

The Brauer group of analytic K3 surfaces

We show that for complex analytic K3 surfaces any torsion class in H^2(X,O_X^*) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to Grothendieck. In our situation, we use twistor spaces to deform a given analytic K3 surface to suitable projective K3 surfaces, and then stable bundles and hyperholomorphy conditions to pass back and forth between the members of the twistor family.

math.AG↗
arXiv2001

Finiteness results for hyperkaehler manifolds

In this short note we prove that the number of deformation types of compact hyperkaehler manifolds with prescribed second cohomology and second Chern class is finite. The proof uses the finiteness result of Kollar and Matsusaka, a formula by Hitchin and Sawon and the surjectivity of the period map.

math.AG↗
arXiv2001

The Kaehler cone of a compact hyperkaehler manifold

This is an attempt towards the understanding of the (birational) Kaehler cone of a compact hyperkaehler manifold in terms of the Beauville-Bogomolov form on its second cohomology. We discuss birational correspondences between hyperkaehler manifolds and their effects on the cohomology. Many of the results are conjectural in as much as they depend on a projectivity criterion for compact hyperkaehler manifolds contained in this paper's predecessor, but in which a serious mistake has oocured. An erratum is given in Sect. 6 and a way to rescue the approach is proposed in Sect. 7.

math.AG↗
arXiv2001

Infinitesimal Variation of Harmonic Forms and Lefschetz Decomposition

This paper studies the infinitesimal variation of the Lefschetz decomposition associated with a compatible sl_2-representation on a graded algebra. This allows to prove that the Jordan-Lefschetz property holds infinitesimally for the Kaehler Lie algebra (introduced by Looijenga and Lunts) of any compact Kaehler manifold. As a second application we describe how the space of harmonic forms changes when a Ricci-flat Kaehler form is deformed infinitesimally.

math.AG↗
arXiv1996

Birational symplectic manifolds and their deformations

The known counterexamples to the global Torelli theorem for higher-dimensional hyperkahler manifolds are provided by birational manifolds. We address the question whether two birational hyperkahler manifolds (i.e. irreducible symplectic) manifolds always define non-separated points in the moduli space of marked manifolds. An affirmative answer is given for the cases of Mukai's elementary transformations and birational correspondences which are isomorphic in codimension two. The techniques are applied to show that the moduli spaces of rank two sheaves on a K3 surface are deformation equivalent to appropriate Hilbert schemes.

alg-geom↗
arXiv1994

Hodge numbers of moduli spaces of stable bundles on K3 surfaces

We show that the Hodge numbers of the moduli space of stable rank two sheaves with primitive determinant on a K3 surface coincide with the Hodge numbers of an appropriate Hilbert scheme of points on the K3 surface. The precise result is: Theorem: Let $X$ be a K3 surface, $L$ a primitive big and nef line bundle and $H$ a generic polarization. If the moduli space of rank two $H$ semi-stable torsion-free sheaves with determiant $L$ and second Chern class $c_2$ has at least dimension 10 then its Hodge numbers coincide with those of the Hilbert scheme of $l:=2c_2-\frac{L^2}{2}-3$ points on $X$.

alg-geom↗
arXiv1992

Stable pairs on curves and surfaces

We describe stability conditions for pairs consisting of a coherent sheaf and a homomorphism to a fixed coherent sheaf on a projective variety. The corresponding moduli spaces are constructed for pairs on curves and surfaces. We consider two examples. The fixed sheaf is the structure sheaf or is a vector bundle on a divisor, i.e. Higgs pairs or framed bundles, resp. (unencoded version)

alg-geom↗