Equivalences of twisted K3 surfaces

Abstract

Abstract. 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.By definition a K3 surface is a compact complex surface X with trivial canonical bundle and vanishing H 1 (X, O X ). As was shown by Kodaira in [23] all K3 surfaces are deformation equivalent. In particular, any K3 surface is diffeomorphic to the four-dimensional manifold M underlying the Fermat quartic in P 3 defined by x 4 0 + x 4 1 + x 4 2 + x 4 3 = 0. Thus, we may think of a K3 surface X as a complex structure I on M . (As it turns out, every complex structure on M does indeed define a K3 surface, see [14].)In the following, we shall fix the orientation on M that is induced by a complex structure and denote by Λ the cohomology H 2 (M, Z) endowed with the intersection pairing. This is an even unimodular lattice of signature (3,19) and hence isomorphic to (−E 8 ) ⊕2 ⊕ U ⊕3 with E 8 the unique even positive definite unimodular lattice of rank eight and U the hyperbolic plane.We shall denote by Λ the lattice given by the full integral cohomology H * (M, Z) (which is concentrated in even degree) endowed with the Mukai pairing ϕ 0 + ϕ 2 + ϕ 4 , ψ 0 + ψ 2 + ψ 4 = ϕ 2 ∧ ψ 2 − ϕ 0 ∧ ψ 4 − ϕ 4 ∧ ψ 0 . In other words, Λ is the direct sum of (H 0 ⊕ H 4 )(M, Z) endowed with the negative intersection pairing and Λ. Hence, Λ ∼ = Λ ⊕ U .An isomorphism between two K3 surfaces X and X ′ given by two complex structures I respectively I ′ on M is a diffeomorphism f ∈ Diff(M ) such that I = f * (I ′ ). Any such diffeomorphism f acts on the cohomology of M and, therefore, induces a lattice automorphism f * : Λ ∼ = Λ.Conversely, one might wonder whether any element g ∈ O(Λ) is of this form. This is essentially true and has been proved by Borcea in [4]. The precise statement is:For any g ∈ O(Λ) there exist two K3 surfaces X = (M, I) and X ′ = (M, I ′ ) and an isomorphism f : X ∼ = X ′ with f * = ±g.(In fact, we can even prescribe the K3 surface X = (M, I), but stated like this the result compares nicely with Theorem 0.1.) The proof of this fact In a next step, we consider a more flexible notion of isomorphisms of K3 surfaces: One says that two K3 surfaces X and X ′ are derived equivalent if there exists a Fourier-Mukai equivalence Φ :is the bounded derived category of the abelian category Coh(X) of coherent sheaves on X. (Usually, derived equivalence is only considered for algebraic K3 surfaces.)Clearly, any isomorphism between X and X ′ given by f ∈ Diff(M ) induces a Fourier-Mukai equivalence Φ := Rf * . By results of Mukai and Orlov one knows how to associate to any Fourier-Mukai equivalence Φ :o...

Cubic fourfolds containing a plane and a quintic del Pezzo surface

“…By [HS05,Rem. 7.10], given any K3 surface S and any nontrivial β ∈ Br(S), there is no equivalence between D b (S, β) and D b (S).…”

Derived equivalences of K3 surfaces and orientation

“…satisfies Φ H * E 0 = (−id H 2 ) ⊕ id H 0 ⊕H 4 . As all orientation preserving Hodge isometries do lift to autoequivalences (see [15,21,31] or [17,Ch. 10]), this seemingly weaker form is equivalent to the original Theorem 2.…”

Generalized Calabi–yau Structures, K3 Surfaces, and B-Fields

“…In the twisted case, T (ϕ) ⊥ does not necessarily contain such an hyperbolic plane. In fact, in [13] we give an explicit example of an Hodge isometry T (ϕ) ∼ = T (ϕ ′ ) which cannot be extended simply due to the fact that the Picard groups are not isometric. We furthermore put forward a version of Cȃldȃraru's conjecture that takes into account not only the transcendental lattice T (X, α B ) ∼ = T (ϕ), but the full Hodge structure defined by the generalized Calabi-Yau structure ϕ = σ + B ∧ σ.…”