Advertisement

Inventiones mathematicae

, Volume 128, Issue 1, pp 45–88 | Cite as

The intrinsic normal cone

  • K. Behrend
  • B. Fantechi

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.

Keywords

Modulus Space Normal Cone Deformation Theory Relative Context Chow Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • K. Behrend
    • 1
  • B. Fantechi
    • 2
  1. 1.University of British Columbia, Mathematics Department, 121–1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada (e-mail: behrend@math.ubc.ca)CA
  2. 2.Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38050 Povo, Italy (e-mail: fantechi@alpha.science.unitn.it)IT

Personalised recommendations