A Glimpse of Deformation Theory

The purpose of these notes, as suggested by the title, is not to provide any sort of comprehensive introduction to deformation theory. Rather, we attempt to convey the main ideas of the theory, with a survey of some applications. We do not explore an exhaustive list of possible topics, nor do we go into details in many proofs. However, in the case of deformations of smooth varieties we do attempt to give a thorough treatment of the theory of first-order infinitesimal deformations, with some hints as to how to generalize to other problems. Because the systematic use of rings with nilpotents is one of the major distinguishing characteristics of scheme theory, we can view deformation theory as a substantial application of scheme theory that is beyond the reach of classical algrebraic geometry. We hope to provide a relatively accessible and motivated introduction to the theory of cohomology of sheaves, in the form of the Cech cohomology arising in deformations of smooth varieties.