Enumeration of Rational Curves Via Torus Actions

  • Maxim Kontsevich
Part of the Progress in Mathematics book series (PM, volume 129)


This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry.


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  1. [AM]
    P. S. Aspinwall, D. R. Morrison, Topological field theory and rational curves, Commun. Math. Phys. 151 (1993), 245–262.MathSciNetMATHCrossRefGoogle Scholar
  2. [BFM]
    P. Baum, W. Fulton, R. MacPherson, Riemann-Roch for singular varieties, Publ. Math. I. H. E. S. 45 (1975), 101–145.MathSciNetMATHGoogle Scholar
  3. [B]
    R. Bott, A residue formula for holomorphic vector fields, Jour. Diff. Geom. 1 (1967), 311–330.Google Scholar
  4. [DM]
    P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus, Publ. Math. I. H. E. S. 36 (1969), 75–110.MathSciNetMATHGoogle Scholar
  5. [GK]
    A. Givental, B. Kim, Quantum cohomology of flag manifolds and Toda lattices, preprint (1993).Google Scholar
  6. [ID]
    C. Itzykson, J.-M. Drouffe, Statistical field theory, Cambridge University Press, 1989.Google Scholar
  7. [K]
    M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys. 147 (1992), 1–23.CrossRefGoogle Scholar
  8. [KM]
    M. Kontsevich, Yu. Manin, Gromov-Witten classes, quantum cohomology and enumerative geometry, MPI preprint and hep-th/9402147 (1994).Google Scholar
  9. M] Yu. Manin, Generating functions in algebraic geometry and sums over trees, this volume.Google Scholar
  10. [P]
    P. Pansu, Chapter VIII, Compactness, Holomorphic curves in symplectic geometry, eds. M. Audin, J. Lafontaine, Progress in Mathematics, vol. 117, Birkhauser, 1994.Google Scholar
  11. [R]
    Z. Ran, Enumerative geometry of singular plane curves, Invent. Math. 97 (1989), 447–465.MATHGoogle Scholar
  12. [RT]
    Y. Ruan, G. Tian, Mathematical theory of quantum cohomology, preprint (1993).Google Scholar
  13. [W]
    E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys in Diff. Geom. 1 (1991), 243–310.Google Scholar
  14. [Y]
    S. T. Yau, ed., Essays on Mirror Manifolds, International Press Co., Hong Kong, 1992.MATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • Maxim Kontsevich
    • 1
    • 2
  1. 1.Dept. of MathematicsUniversity of CaliforniaBerkeleyUSA
  2. 2.Max-Planck-Institut Für MathematikBonn and University of CaliforniaBerkeleyUSA

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