Equivalences of twisted K3 surfaces

By D. Huybrechts, P. Stellari. 2004

130 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6191a6901cbb09c84fc444a0986a48af9ac69136

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Citations in the corpus

Twisted Fourier–Mukai functors (2006, methodology): “This was first proved by Căldăraru ([3, 4]) and then generalized in [12, 13, 17] and [9, 10].”
Moduli spaces of twisted sheaves on a projective variety (2004, background): “ttice. Calda˘raru conjectured that a similar result also holds for the derived category of twisted sheaves. Recently this conjecture was modiﬁed and proved by Huybrechts and Stellari, if ρ(X) ≥ 12 in [H-St]. As is well-known, twisted sheaves also appear if we consider a projective bundle over X. In ”
A categorical invariant for cubic threefolds (2009, unknown): “ the second part of the statement, assume there exists an equivalence Φ : Db(S1,α1) −→∼ Db(S2,α2) inducing, as before, a Hodge isometry Φ∗ : T(S1,α1) −→∼ T(S 2,α2). Take a B-ﬁeld lift Bi for αi. (See [14] for more details.) We want to show that there is a Hodge isometry f : T(S1) → T(S2) making the ”
Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces (2018, background): “ple 3.18. Let (S,α) be a twisted K3 surface. Then the total cohomology H∗(S,Z) is endowed with a Mukai pairing and a weight-2 Hodge structure which depends on a lift to H2(S,Q) of α(see, for example, [HS05]). This lattice with this Hodge structure is called the Mukai lattice and it is denoted by He(”
Lectures on K3 Surfaces (2016, background): “However, then Db(X) ' D(X0, αn) for infinitely many Brauer class, which in [368] is excluded by lifting to characteristic zero and then using a finiteness result in [269].”
On the monodromy of moduli spaces of sheaves on K3 surfaces II (2003, background): “s also covariant. Conjecturally, any auto-equivalence, of the derived category of a K3 surface, is sent by the covariance character to 0 (see [Sz, HLOY]). Many cases of this conjecture were proven in [HS] Proposition 5.5. On the other hand, the Duality Hodge isometry (29) is contravariant, cov(D) = ”
Derived equivalences of K3 surfaces and orientation (2007, background): “As all orientation preserving Hodge isometries do lift to autoequivalences (see [15, 21, 31] or [17, Ch.”
Derived Categories of Cubic Fourfolds (2008, methodology): “By [ HS1 , Proposition 1.2], there exists a linear map (the B-twisted Chern character)”
MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations (2013, background): “In [HS05], the authors define a weight-2 Hodge structure on the whole cohomology H∗(X,α,Z) with H(X,α,C) := e0 ·H(X,C).”
Projectivity and birational geometry of Bridgeland moduli spaces (2012, unknown): “A twisted Chern character for elements of Coh(X,α) has been defined in [HS05].”
Fano varieties of cubic fourfolds containing a plane (2009, background): “1]) and T (S,B) can be realized as a (non-primitive) sublattice of T (S) of index d (see, for example, [22]).”
Inducing stability conditions (2007, background): “Notice that for K3 surfaces a Derived Torelli Theorem for K3 surfaces is already present in the literature ([16, 17, 19, 32, 33]).”