Deformation theory from the point of view of fibered categories

Abstract

We give an exposition of the formal aspects of deformation theory in the language of fibered categories, instead of the more traditional one of functors. The main concepts are that of tangent space to a deformation problem, obstruction theory, versal and universal formal deformations. We include proofs of two key results: a versione of Schlessinger's Theorem in this context, and the Ran--Kawamata vanishing theorem for obstructions. We accompany this with a detailed analysis of three important cases: smooth varieties, local complete intersection subschemes and coherent sheaves.

arXiv commentcorrected several typos and made some minor improvements to the exposition

References in corpus (1)

• Revêtements étales et groupe fondamental (SGA 1)

Cited by in corpus (6)

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• Virtual classes of Artin stacks
• On deformation spaces of toric singularities and on singularities of K-moduli of Fano varieties
• A support theorem for the Hitchin fibration: the case of $G{L}_n$ and $K_C$
• Curves with prescribed symmetry and associated representations of mapping class groups
• Unobstructedness of deformations of Calabi-Yau varieties with a line bundle