Inventiones mathematicae

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

The intrinsic normal cone

  • K. Behrend
  • B. Fantechi


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.


Modulus Space Normal Cone Deformation Theory Relative Context Chow Group 
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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:
  2. 2.Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38050 Povo, Italy (e-mail:

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