The intrinsic normal cone

Abstract

Let $X$ be an algebraic stack in the sense of Deligne-Mumford. We construct a purely $0$ -dimensional algebraic stack over $X$ (in the sense of Artin), the intrinsic normal cone ${\frak C}_X$ . The notion of (perfect) obstruction theory for $X$ is introduced, and it is shown how to construct, given a perfect obstruction theory for $X$ , a pure-dimensional virtual fundamental class in the Chow group of $X$ . We then prove some properties of such classes, both in the absolute and in the relative context. Via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory, and sometimes a perfect one.