Pour une variété complexe projective il est possible de construire les invariants de Gromov-Witten avec des méthodes algébriques ou symplectiques. Utilisant l’approche algébrique de Behrend et Fantechi et l’approche symplectique de l’auteur, on prouve l’équivalence des deux points de vue.
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
@article{AIF_1999__49_6_1743_0, author = {Siebert, Bernd}, title = {Algebraic and symplectic {Gromov-Witten} invariants coincide}, journal = {Annales de l'Institut Fourier}, pages = {1743--1795}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {49}, number = {6}, year = {1999}, doi = {10.5802/aif.1737}, mrnumber = {2001f:14097}, zbl = {0970.14030}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1737/} }
TY - JOUR AU - Siebert, Bernd TI - Algebraic and symplectic Gromov-Witten invariants coincide JO - Annales de l'Institut Fourier PY - 1999 SP - 1743 EP - 1795 VL - 49 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.1737/ DO - 10.5802/aif.1737 LA - en ID - AIF_1999__49_6_1743_0 ER -
%0 Journal Article %A Siebert, Bernd %T Algebraic and symplectic Gromov-Witten invariants coincide %J Annales de l'Institut Fourier %D 1999 %P 1743-1795 %V 49 %N 6 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.1737/ %R 10.5802/aif.1737 %G en %F AIF_1999__49_6_1743_0
Siebert, Bernd. Algebraic and symplectic Gromov-Witten invariants coincide. Annales de l'Institut Fourier, Tome 49 (1999) no. 6, pp. 1743-1795. doi : 10.5802/aif.1737. http://archive.numdam.org/articles/10.5802/aif.1737/
[Be] GW-invariants in algebraic geometry, Inv. Math., 127 (1997), 601-617. | MR | Zbl
,[BeFa] The intrinsic normal cone, Inv. Math., 128 (1997), 45-88. | MR | Zbl
, ,[BeMa] Stacks of stable maps and Gromov-Witten invariants, Duke. Math. Journ., 85 (1996), 1-60. | MR | Zbl
, ,[BiKo] Lokale Modulräume in der analytischen Geometrie I, II, Vieweg 1987. | Zbl
, ,[Do] Le problème des modules locaux pour les espaces ℂ-analytiques compacts, Ann. Sci. École Norm. Sup. (4), 7 (1974), 569-602. | Numdam | MR | Zbl
,[Fi] Complex analytic geometry, Lecture Notes Math., 538, Springer, 1976. | MR | Zbl
,[Fl] Über Deformationen holomorpher Abbildungen, Habilitationsschrift, Univ. Osnabrück, 1978.
,[FkOn] Arnold conjecture and Gromov-Witten invariant, Warwick preprint 29/1996.
, ,[Fu] Intersection theory, Springer, 1984. | MR | Zbl
,[FuPa] Notes on stable maps and quantum cohomology, to appear in the proceedings of the Algebraic Geometry Conference, Santa Cruz, 1995. | Zbl
, ,[Ha] Residues and duality, Lecture Notes Math., 20, Springer, 1966. | Zbl
,[Il] Complexe cotagent et déformations I, Lecture Notes Math., 239, Springer, 1971. | MR | Zbl
,[Iv] Cohomology of Sheaves, Springer, 1986. | MR | Zbl
,[Ka] The signature theorem for V-manifolds, Topology, 17 (1978), 75-83. | MR | Zbl
,[Kn] The projectivity of the moduli space of stable curves, II: The stacks Mg, n, Math. Scand., 52 (1983), 161-199. | MR | Zbl
,[KpKp] Holomorphic functions of several variables, de Gruyter, 1983.
, ,[LiTi1] Virtual moduli cycles and GW-invariants of algebraic varieties, Journal Amer. Math. Soc., 11 (1998), 119-174. | MR | Zbl
, ,[LiTi2] Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, preprint alg-geom/9608032. | Zbl
, ,[LiTi3] Comparison of the algebraic and the symplectic Gromov-Witten invariants, preprint alg-geom/9712035. | Zbl
, ,[Lp] Notes on Derived Categories and Derived Functors, preprint, available from http://www.math.purdue.edu/~lipman.
,[Po] Théorème de Douady au-dessus de S, Ann. Scuola Norm. Sup. Pisa, 23 (1969), 451-459. | Numdam | MR | Zbl
,[RaRuVe] Dualité relative en géométrie analytique complexe, Invent. Math., 13 (1971), 261-283. | MR | Zbl
, , ,[Ru1] Topological sigma model and Donaldson type invariants in Gromov theory, Duke Math. Journ., 83 (1996), 461-500. | MR | Zbl
,[Ru2] Virtual neighborhoods and pseudo-holomorphic curves, preprint alg-geom/ 9611021. | Zbl
,[RuTi1] A mathematical theory of quantum cohomology, Journ. Diff. Geom., 42 (1995), 259-367. | MR | Zbl
, ,[RuTi2] Higher genus symplectic invariants and sigma model coupled with gravity, Inv. Math., 130 (1997), 455-516. | MR | Zbl
, ,[Sa] The Gauss-Bonnet theorem for V-manifolds, Journ. Math. Soc. Japan, 9 (1957), 164-492. | MR | Zbl
,[Se] Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble, 6 (1956), 1-42. | Numdam | MR | Zbl
,[Si1] Gromov-Witten invariants for general symplectic manifolds, preprint dg-ga/9608032, revised 12/97.
,[Si2] Global normal cones, virtual fundamental classes and Fulton's canonical classes, preprint 2/1997.
,[Si3] 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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