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 prob-lem, obstruction theory, versal and universal formal deformations. We include proofs of two key results: a version 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