Abstract
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.
Highlights
Stability conditions on triangulated categories were introduced by Bridgeland in [Bri07] and further studied by Kontsevich and Soibelman in [KS08]; originally based on work by Douglas [Dou02] in string theory, they have found many applications to algebraic geometry via wall-crossing
Theorem 4.13. — Let X → S be a morphism schemes which are quasi-compact with affine diagonal, where X is noetherian of finite Krull dimension and S admits an ample line bundle L, and let D ⊂ Db(X) be an S-linear strong semiorthogonal component whose projection functor is of finite cohomological amplitude
Stability conditions over a higher-dimensional base. In this part of the paper, we introduce a notion of stability conditions for a category D over a higher-dimensional base S; its key property will be that it comes equipped with relative moduli spaces
Summary
Stability conditions on triangulated categories were introduced by Bridgeland in [Bri07] and further studied by Kontsevich and Soibelman in [KS08]; originally based on work by Douglas [Dou02] in string theory, they have found many applications to algebraic geometry via wall-crossing. The first application concerns moduli spaces Mσ (Ku(X), v) of σ -stable objects in Ku(X) with Mukai vector v, for σ a Bridgeland stability condition in the distinguished connected component Stab†(Ku(X)), generalizing a series of results for K3 surfaces [Bea, Muk, Muk, O’G97, Huy, Yos, Tod, BM14b]. — For any pair (a, b) of coprime integers, there is a unirational locally complete 20-dimensional family, over an open subset of the moduli space of cubic fourfolds, of polarized smooth projective irreducible holomorphic symplectic manifolds of dimension 2n + 2, where n = a2 − ab + b2. Even in the case of cubic fourfolds, there are other examples of families of polarized hyperkähler manifolds associated to some geometric construction which should be possible to interpret as relative moduli spaces. Some results work is greater generality and we will explain this by recalling the precise assumptions in each section
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