Stability conditions in families

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.

146 pages

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Cited by in corpus (26)

• Stability conditions on Kuznetsov components
• The generalized Franchetta conjecture for some hyper-Kähler varieties, II
• Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties
• The desingularization of the theta divisor of a cubic threefold as a moduli space
• Birational geometry of the intermediate Jacobian fibration of a cubic fourfold
• Stability conditions on threefolds with nef tangent bundles
• A note on the Kuznetsov component of the Veronese double cone
• Some remarks on Fano threefolds of index two and stability conditions
• Derived categories of Quot schemes of locally free quotients
• Categorical cones and quadratic homological projective duality
• Bridgeland stability conditions on some threefolds of general type
• Symbols and equivariant birational geometry in small dimensions
• Stability conditions on product varieties
• Automorphism groups of cubic fourfolds and K3 categories
• On the period of Lehn, Lehn, Sorger, and van Straten's symplectic eightfold
• Bridgeland Moduli spaces for Gushel-Mukai threefolds and Kuznetsov's Fano threefold conjecture
• Rational cubic fourfolds in Hassett divisors
• Some Quot schemes in tilted hearts and moduli spaces of stable pairs
• Weight functions, tilts, and stability conditions
• MMP Via Wall-crossing for Moduli Spaces of Stable Sheaves on an Enriques surface
• Serre functors and dimensions of residual categories
• Moduli of Bridgeland semistable holomorphic triples
• Ulrich bundles on cubic fourfolds
• Fullness of exceptional collections via stability conditions -- A case study: the quadric threefold
• On lattice polarizable cubic fourfolds
• Virtual classes of ${\mathbb{G}}_{\text{m}}$-gerbes