Daniel Huybrechts

Compact hyperkähler manifolds: basic results

Compact hyperkähler manifolds, or irreducible symplectic manifolds as they will be frequently called in these notes, are higher-dimensional analogues of K3 surfaces. That they indeed share many of the well-known properties of K3 surfaces this paper intends to show. To any K3 surface X one naturally associates its period consisting of the weight-two Hodge structure on H2(X,Z) together with the intersection pairing. The period describes and determines a K3 surface in the following sense:

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.

Derived equivalences of K3 surfaces and orientation

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.

STABILITY CONDITIONS FOR GENERIC K3 CATEGORIES

A K3 category is by definition a Calabi-Yau category of dimension two. Geo- metrically 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.

Compact Hyperkähler Manifolds

Ricci-flat manifolds have been studied for many years and the interest in various aspects of their theory seems still to be growing. Their rich geometry has been explored with techniques from different branches of mathematics and the interplay of analysis, arithmetic, and geometry makes their theory highly attractive. At the same time, these manifolds play a prominent role in string theory and, in particular, in mirror symmetry. Physicists have come up with interesting and difficult mathematical questions and have suggested completely new directions that should be pursued.

The K3 category of a cubic fourfold

Smooth cubic fourfolds are linked to K3 surfaces via their Hodge structures, due to work of Hassett, and via Kuznetsovs K3 category A. The relation between these two viewpoints has recently been elucidated by Addington and Thomas. In this paper, both aspects are studied further and extended to twisted K3 surfaces, which in particular allows us to determine the group of autoequivalences of A for the general cubic fourfold. Furthermore, we prove finiteness results for cubics with equivalent K3 categories and study periods of cubics in terms of generalized K3 surfaces.

The Brauer group of analytic K3 surfaces

We show that for complex analytic K3 surfaces any torsion class in H2(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. In analogy to the isomorphism Pic(X) ∼= H1(X,O∗ X), Grothendieck investigated in [8] the possibility of interpreting classes in H2(X,O∗ X) as geometric objects. He observed that the Brauer group Br(X), parameterizing equivalence classes of sheaves of Azumaya algebras on X, naturally injects into H2(X,O∗ X). It is not difficult to see that Br(X) ⊂ H2(X,O∗ X) is contained in the torsion part of H2(X,O∗ X) and Grothendieck asked: Is the natural injection Br(X ) ⊂ H (X ,O∗ X )tor an isomorphism? This question is of interest in various geometric categories, e.g. X might be a scheme, a complex space, a complex manifold, etc. It is also related to more recent developments in the application of complex algebraic geometry to conformal field theory. Certain elements in H2(X,O∗ X) have been interpreted as so-called B-fields, and those are used to construct super conformal field theories associated to Ricci-flat manifolds. Thus, understanding the geometric meaning of the cohomological Brauer group Br′(X) := H2(X,O∗ X)tor is also of interest for the mathematical interpretation of string theory and mirror symmetry. An affirmative answer to Grothendieck’s question has been given only in very few special cases: • If X is a complex curve, then H2(X,O∗ X) = 0. Hence, Br(X) = Br ′(X) = H2(X,O∗ X) = 0 (see [8, Cor.2.2] for the general case of a curve). • For smooth algebraic surfaces the surjectivity has been proved by Grothendieck [8, Cor.2.2] and for normal algebraic surfaces a proof was given more recently by Schroer [14]. • Hoobler [9] and Berkovich [3] gave an affirmative answer for abelian varieties of any dimension and Elencwajg and Narasimhan gave another proof for complex tori [6].

Chow groups of K3 surfaces and spherical objects

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

Curves and cycles on K3 surfaces

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FORMAL DEFORMATIONS AND THEIR CATEGORICAL GENERAL FIBRE

, this is compatible with the Bloch-Beilinson conjecture. Besides the work of Beauville and Voisin, Lazarfelds result on Brill-Noether theory for curves in K3 surfaces and the deformation theory developed with Macri and Stellari in arXiv:0710.1645 are central for the discussion.