The global Torelli theorem: classical, derived, twisted

Abstract

These notes survey work on various aspects and generalizations of the Global Torelli Theorem for K3 surfaces done over the last ten years. The classical Global Torelli Theorem was proved a long time ago (see [9,39,52,57]), but the interest in similar questions has been revived by the new approach to K3 surfaces suggested by mirror symmetry.Kontsevich proposed to view mirror symmetry as an equivalence between the bounded derived category of coherent sheaves on a Calabi-Yau manifold and the derived Fukaya category of its mirror dual. For an algebraic geometer the bounded derived category of coherent sheaves on a variety is a familiar object, but to view it as an interesting invariant of the variety rather than a technical tool to deal with cohomology is rather surprising. Due to results of Mukai, Orlov, and Polishchuk, the bounded derived category of coherent sheaves on an abelian variety is completely understood, i.e. we know when two abelian varieties yield equivalent derived categories and what the group of autoequivalences looks like.Independently of their importance in mirror symmetry, K3 surfaces form the next most simple class of Calabi-Yau manifolds and one would like to study them from a derived category point of view, too. The program has been started already by Mukai in [43] and the break-through came with Orlov's article [48]. But this was still not the end. For many reasons (mirror symmetry considerations, existence of non-fine moduli spaces, geometric interpretation of conformal field theories, etc.) one would like to incorporate B-fields or, closely related, Brauer classes in the picture. These notes will mostly concentrate on aspects that are related to generalizations of the Global Torelli Theorem in this direction. The following list of topics gives an idea of what shall be discussed:• Hitchin's generalized Calabi-Yau structures.• The period of a twisted K3 surface.• Brauer group and B-fields.• Derived categories of twisted sheaves.This survey contains essentially no proofs. I have tried to emphasize ideas and refer to the literature for details. Some of the results can very naturally be viewed in terms of moduli spaces of generalized K3 surfaces and the action of the mirror symmetry group, but I have decided to neglect these aspects almost completely. Although mirror symmetry has shaped our way of thinking about derived categories, the symplectic side of the theory will not be touched upon.

The K3 category of a cubic fourfold

“…Most of the theory for untwisted K3 surfaces is due to Mukai [Muk87] and Orlov [Orl97], whereas the basic theory of twisted K3 surfaces was developed in [HS05, HS06]. See also [Huy06, Huy09] for surveys and further references. Originally, the generalization to twisted K3 surfaces was motivated by the existence of non-fine moduli spaces [Căl00].…”

Moduli spaces of twisted K3 surfaces and cubic fourfolds

“…For references, see [14,13]. Recall that the Brauer group Br(X ) of a scheme X is the group of sheaves of Azumaya algebras modulo Morita equivalence, with multiplication given by the tensor product.…”

Motives of isogenous K3 surfaces

“…[ϕ] = n ch(E ⊗n ) · td(S × S ′ ) (2,2) ∈ H 2 (S, Q) ⊗ H 2 (S ′ , Q), 3 We refer to [8,10,11] which is clearly an algebraic class.…”

“…Note that by Witt's theorem, there exists a Hodge isometry H 2 (S, Q) ≃ H 2 (S ′ , Q) if and only if there exists a Hodge isometry T (S) ⊗ Q ≃ T (S ′ ) ⊗ Q. For integral coefficients this fails, which results in two global Torelli theorems, the classical and the derived, 1 see [9,11,12] for references.…”