Algebraic and symplectic Gromov-Witten invariants coincide

Bernd Siebert

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 6, page 1743-1795
  • ISSN: 0373-0956

Abstract

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For a complex projective manifold Gromov-Witten invariants can be constructed either algebraically or symplectically. Using the versions of Gromov-Witten theory by Behrend and Fantechi on the algebraic side and by the author on the symplectic side, we prove that both points of view are equivalent

How to cite

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Siebert, Bernd. "Algebraic and symplectic Gromov-Witten invariants coincide." Annales de l'institut Fourier 49.6 (1999): 1743-1795. <http://eudml.org/doc/75401>.

@article{Siebert1999,
abstract = {For a complex projective manifold Gromov-Witten invariants can be constructed either algebraically or symplectically. Using the versions of Gromov-Witten theory by Behrend and Fantechi on the algebraic side and by the author on the symplectic side, we prove that both points of view are equivalent},
author = {Siebert, Bernd},
journal = {Annales de l'institut Fourier},
keywords = {Gromov-Witten invariants; virtual fundamental class; Grothendieck duality; derived category; moduli space; homology class},
language = {eng},
number = {6},
pages = {1743-1795},
publisher = {Association des Annales de l'Institut Fourier},
title = {Algebraic and symplectic Gromov-Witten invariants coincide},
url = {http://eudml.org/doc/75401},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Siebert, Bernd
TI - Algebraic and symplectic Gromov-Witten invariants coincide
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 6
SP - 1743
EP - 1795
AB - For a complex projective manifold Gromov-Witten invariants can be constructed either algebraically or symplectically. Using the versions of Gromov-Witten theory by Behrend and Fantechi on the algebraic side and by the author on the symplectic side, we prove that both points of view are equivalent
LA - eng
KW - Gromov-Witten invariants; virtual fundamental class; Grothendieck duality; derived category; moduli space; homology class
UR - http://eudml.org/doc/75401
ER -

References

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  1. [Be] K. BEHREND, GW-invariants in algebraic geometry, Inv. Math., 127 (1997), 601-617. Zbl0909.14007MR98i:14015
  2. [BeFa] K. BEHREND, B. FANTECHI, The intrinsic normal cone, Inv. Math., 128 (1997), 45-88. Zbl0909.14006MR98e:14022
  3. [BeMa] K. BEHREND, Y. MANIN, Stacks of stable maps and Gromov-Witten invariants, Duke. Math. Journ., 85 (1996), 1-60. Zbl0872.14019MR98i:14014
  4. [BiKo] J. BINGENER, S. KOSAREW, Lokale Modulräume in der analytischen Geometrie I, II, Vieweg 1987. Zbl0644.32001
  5. [Do] A. DOUADY, Le problème des modules locaux pour les espaces ℂ-analytiques compacts, Ann. Sci. École Norm. Sup. (4), 7 (1974), 569-602. Zbl0313.32036MR52 #3611
  6. [Fi] G. FISCHER, Complex analytic geometry, Lecture Notes Math., 538, Springer, 1976. Zbl0343.32002MR55 #3291
  7. [Fl] H. FLENNER, Über Deformationen holomorpher Abbildungen, Habilitationsschrift, Univ. Osnabrück, 1978. 
  8. [FkOn] K. FUKAYA, K. ONO, Arnold conjecture and Gromov-Witten invariant, Warwick preprint 29/1996. 
  9. [Fu] W. FULTON, Intersection theory, Springer, 1984. Zbl0541.14005MR85k:14004
  10. [FuPa] W. FULTON, R. PANDHARIPANDE, Notes on stable maps and quantum cohomology, to appear in the proceedings of the Algebraic Geometry Conference, Santa Cruz, 1995. Zbl0898.14018
  11. [Ha] R. HARTSHORNE, Residues and duality, Lecture Notes Math., 20, Springer, 1966. Zbl0212.26101
  12. [Il] L. ILLUSIE, Complexe cotagent et déformations I, Lecture Notes Math., 239, Springer, 1971. Zbl0224.13014MR58 #10886a
  13. [Iv] B. IVERSEN, Cohomology of Sheaves, Springer, 1986. Zbl0559.55001MR87m:14013
  14. [Ka] T. KAWASAKI, The signature theorem for V-manifolds, Topology, 17 (1978), 75-83. Zbl0392.58009MR57 #14072
  15. [Kn] F. KNUDSON, The projectivity of the moduli space of stable curves, II: The stacks Mg, n, Math. Scand., 52 (1983), 161-199. Zbl0544.14020MR85d:14038a
  16. [KpKp] B. KAUP, L. KAUP, Holomorphic functions of several variables, de Gruyter, 1983. 
  17. [LiTi1] J. LI, G. TIAN, Virtual moduli cycles and GW-invariants of algebraic varieties, Journal Amer. Math. Soc., 11 (1998), 119-174. Zbl0912.14004MR99d:14011
  18. [LiTi2] J. LI, G. TIAN, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, preprint alg-geom/9608032. Zbl0978.53136
  19. [LiTi3] J. LI, G. TIAN, Comparison of the algebraic and the symplectic Gromov-Witten invariants, preprint alg-geom/9712035. Zbl0983.53061
  20. [Lp] J. LIPMAN, Notes on Derived Categories and Derived Functors, preprint, available from http://www.math.purdue.edu/~lipman. 
  21. [Po] G. POURCIN, Théorème de Douady au-dessus de S, Ann. Scuola Norm. Sup. Pisa, 23 (1969), 451-459. Zbl0186.14003MR41 #2053
  22. [RaRuVe] J. P. RAMIS, G. RUGET, J.-L. VERDIER, Dualité relative en géométrie analytique complexe, Invent. Math., 13 (1971), 261-283. Zbl0218.14010MR46 #7553
  23. [Ru1] Y. RUAN, Topological sigma model and Donaldson type invariants in Gromov theory, Duke Math. Journ., 83 (1996), 461-500. Zbl0864.53032MR97d:58042
  24. [Ru2] Y. RUAN, Virtual neighborhoods and pseudo-holomorphic curves, preprint alg-geom/ 9611021. Zbl0967.53055
  25. [RuTi1] Y. RUAN, G. TIAN, A mathematical theory of quantum cohomology, Journ. Diff. Geom., 42 (1995), 259-367. Zbl0860.58005MR96m:58033
  26. [RuTi2] Y. RUAN, G. TIAN, Higher genus symplectic invariants and sigma model coupled with gravity, Inv. Math., 130 (1997), 455-516. Zbl0904.58066MR99d:58030
  27. [Sa] I. SATAKE, The Gauss-Bonnet theorem for V-manifolds, Journ. Math. Soc. Japan, 9 (1957), 164-492. Zbl0080.37403MR20 #2022
  28. [Se] J.-P. SERRE, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble, 6 (1956), 1-42. Zbl0075.30401MR18,511a
  29. [Si1] B. SIEBERT, Gromov-Witten invariants for general symplectic manifolds, preprint dg-ga/9608032, revised 12/97. 
  30. [Si2] B. SIEBERT, Global normal cones, virtual fundamental classes and Fulton's canonical classes, preprint 2/1997. 
  31. [Si3] B. SIEBERT, Symplectic Gromov-Witten invariants, to appear in New trends in Algebraic Geometry, F. Catanese, K. Hulek, C. Peters, M. Reid (eds.), Cambridge Univ. Press, 1998. 

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