Deformation-obstruction theory for complexes via Atiyah and Kodaira--Spencer classes

Abstract

We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that it is a product of Atiyah and Kodaira--Spencer classes. This allows us to obtain deformation-invariant virtual cycles on moduli spaces of objects of the derived category on threefolds.

Corrected flatness problem pointed out by F Langholf and P Bousseau. Updated references. 23 pages

References in corpus (4)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Curve counting via stable pairs in the derived category
• Donaldson-Thomas invariants via microlocal geometry
• Derived equivalences of K3 surfaces and orientation

Cited by in corpus (8)

• Curve counting via stable pairs in the derived category
• Stable pairs and BPS invariants
• Derived equivalences of K3 surfaces and orientation
• Hilbert schemes and stable pairs: GIT and derived category wall crossings
• Generalized Donaldson-Thomas theory over fields K $\ne$ C
• Curve counting theories via stable objects I. DT/PT correspondence
• On the deformation theory of pair (X, E)
• Derived Kodaira Spencer map, Cosection lemma, and semiregularity