Projectivity of Kaehler manifolds - Kodaira's problem (after C. Voisin)

In this expository paper (Bourbaki talk) we survey results of Claire Voisin showing that there exist compact Kaehler manifolds which are not homeomorphic to any projective manifold.Comment: latex 19 page

Finiteness of polarized K3 surfaces and hyperk\"ahler manifolds

In the moduli space of polarized varieties the same unpolarized variety can occur multiple times However, for K3 surfaces, compact hyperk\"ahler manifolds, and abelian varieties the number is finite. This may be viewed as a consequence of the Kawamata-Morrison cone conjecture. In this note we provide a proof of this finiteness not relying on the cone conjecture and, in fact, not even on the global Torelli theorem. Instead, it uses the geometry of the moduli space of polarized varieties to conclude the finiteness by means of Baily-Borel type arguments. We also address related questions concerning finiteness in twistor families associated with polarized K3 surfaces of CM type.Comment: 19 pages, minor corrections, Condition (2.1) and Lemma 2.5 corrected (application in Proposition 2.8 not affected

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.Comment: 24 pages. This final version of the paper, to appear in Int.J.Math., contains further comments on N=(2,2) susy, the definition of twisted Picard and transcendental lattice, and a review of Caldararu's conjecture on the equivalence of twisted derived categorie

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 Kuznetsov's 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.Comment: 40 pages, minor corrections, comments on the order of the Brauer classes added, further corrections, to appear in Compositi

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.Comment: A section is added in which the Mukai lattice is used to explain better some of the formulae. This is the final versio

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 \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:0710.1645 are central for the discussion.Comment: 18 pages, minor modifcations and corrections, to appear in JEM

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.Comment: This is a major revision of the paper. All results based on the projectivity criterion for hyperkahler manifolds have a complete proof now (see the Erratum in alg-geom/9705025). The sections on the Beauville-Bogomolov form are omitted in this version. Further corrections are made in Sect. 1 and

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.Comment: 8 page

Hodge theory of cubic fourfolds, their Fano varieties, and associated K3 categories

These are notes of lectures given at the school `Birational Geometry of Hypersurfaces' in Gargnano in March 2018. The main goal was to discuss the Hodge structures that come naturally associated with a cubic fourfold. The emphasis is on the Hodge and lattice theoretic aspects with many technical details worked out explicitly. More geometric or derived results are only hinted at.Comment: Comments are welcome! This minor revision incorporates suggestions by the referee and corrects typo

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.Comment: Revised version: Lemma in Sect. 3 corrected, typos corrected, Corollary in Sect. 4 adde