We describe a connected component of the space of stability conditions on abelian threefolds, and on Calabi-Yau threefolds obtained as (the crepant resolution of) a finite quotient of an abelian threefold. Our proof includes the following essential steps:1. We simultaneously strengthen a conjecture by the first two authors and Toda, and prove that it follows from a more natural and seemingly weaker statement. This conjecture is a Bogomolov-Gieseker type inequality involv- ing the third Chern character of "tilt-stable" two-term complexes on smooth projective threefolds; we extend it from complexes of tilt-slope zero to arbitrary tilt-slope. 2. We show that this stronger conjecture implies the so-called support property of Bridgeland stability conditions, and the existence of an explicit open subset of the space of stability conditions. 3. We prove our conjecture for abelian threefolds, thereby reproving and generalizing a result by Maciocia and Piyaratne.Important in our approach is a more systematic understanding on the behaviour of quadratic inequalities for semistable objects under wall-crossing, closely related to the support property.

We introduce a general method to induce Bridgeland stability conditions on semiorthogonal decompositions. In particular, we prove the existence of Bridgeland stability conditions on the Kuznetsov component of the derived category of many Fano threefolds (including all but one deformation type of Picard rank one), and of cubic fourfolds. As an application, in the appendix, written jointly with Xiaolei Zhao, we give a variant of the proof of the Torelli theorem for cubic fourfolds by Huybrechts and Rennemo.

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...

We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class.