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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties
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by Jun Li and Gang Tian
J. Amer. Math. Soc. 11 (1998), 119-174


In this paper, we define the virtual moduli cycle of moduli spaces with perfect tangent-obstruction theory. The two interesting moduli spaces of this type are moduli spaces of vector bundles over surfaces and moduli spaces of stable morphisms from curves to projective varieties. As an application, we define the Gromov-Witten invariants of smooth projective varieties and prove all its basic properties.
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Bibliographic Information
  • Jun Li
  • Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
  • MR Author ID: 1093262
  • Email:
  • Gang Tian
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • MR Author ID: 220655
  • Email:
  • Received by editor(s): September 25, 1996
  • Received by editor(s) in revised form: July 30, 1997
  • Additional Notes: Both authors were supported in part by NSF grants, and the first author was also supported by an A. Sloan fellowship and Terman fellowship.
  • © Copyright 1998 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 11 (1998), 119-174
  • MSC (1991): Primary 14D20
  • DOI:
  • MathSciNet review: 1467172