Algebraic and symplectic Gromov-Witten invariants coincide
Siebert, Bernd
Annales de l'Institut Fourier, Tome 49 (1999), p. 1743-1795 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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},
     volume = {49},
     year = {1999},
     pages = {1743-1795},
     doi = {10.5802/aif.1737},
     mrnumber = {2001f:14097},
     zbl = {0970.14030},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1999__49_6_1743_0}
}

Siebert, Bernd. Algebraic and symplectic Gromov-Witten invariants coincide. Annales de l'Institut Fourier, Tome 49 (1999) pp. 1743-1795. doi : 10.5802/aif.1737. http://gdmltest.u-ga.fr/item/AIF_1999__49_6_1743_0/

[Be] K. Behrend, GW-invariants in algebraic geometry, Inv. Math., 127 (1997), 601-617. | MR 98i:14015 | Zbl 0909.14007

[BeFa] K. Behrend, B. Fantechi, The intrinsic normal cone, Inv. Math., 128 (1997), 45-88. | MR 98e:14022 | Zbl 0909.14006

[BeMa] K. Behrend, Y. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke. Math. Journ., 85 (1996), 1-60. | MR 98i:14014 | Zbl 0872.14019

[BiKo] J. Bingener, S. Kosarew, Lokale Modulräume in der analytischen Geometrie I, II, Vieweg 1987. | Zbl 0644.32001

[Do] A. Douady, Le problème des modules locaux pour les espaces ℂ-analytiques compacts, Ann. Sci. École Norm. Sup. (4), 7 (1974), 569-602. | Numdam | MR 52 #3611 | Zbl 0313.32036

[Fi] G. Fischer, Complex analytic geometry, Lecture Notes Math., 538, Springer, 1976. | MR 55 #3291 | Zbl 0343.32002

[Fl] H. Flenner, Über Deformationen holomorpher Abbildungen, Habilitationsschrift, Univ. Osnabrück, 1978.

[FkOn] K. Fukaya, K. Ono, Arnold conjecture and Gromov-Witten invariant, Warwick preprint 29/1996.

[Fu] W. Fulton, Intersection theory, Springer, 1984. | MR 85k:14004 | Zbl 0541.14005

[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. | Zbl 0898.14018

[Ha] R. Hartshorne, Residues and duality, Lecture Notes Math., 20, Springer, 1966. | Zbl 0212.26101

[Il] L. Illusie, Complexe cotagent et déformations I, Lecture Notes Math., 239, Springer, 1971. | MR 58 #10886a | Zbl 0224.13014

[Iv] B. Iversen, Cohomology of Sheaves, Springer, 1986. | MR 87m:14013 | Zbl 0559.55001

[Ka] T. Kawasaki, The signature theorem for V-manifolds, Topology, 17 (1978), 75-83. | MR 57 #14072 | Zbl 0392.58009

[Kn] F. Knudson, The projectivity of the moduli space of stable curves, II: The stacks Mg, n, Math. Scand., 52 (1983), 161-199. | MR 85d:14038a | Zbl 0544.14020

[KpKp] B. Kaup, L. Kaup, Holomorphic functions of several variables, de Gruyter, 1983.

[LiTi1] J. Li, G. Tian, Virtual moduli cycles and GW-invariants of algebraic varieties, Journal Amer. Math. Soc., 11 (1998), 119-174. | MR 99d:14011 | Zbl 0912.14004

[LiTi2] J. Li, G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, preprint alg-geom/9608032. | Zbl 0978.53136

[LiTi3] J. Li, G. Tian, Comparison of the algebraic and the symplectic Gromov-Witten invariants, preprint alg-geom/9712035. | Zbl 0983.53061

[Lp] J. Lipman, Notes on Derived Categories and Derived Functors, preprint, available from http://www.math.purdue.edu/~lipman.

[Po] G. Pourcin, Théorème de Douady au-dessus de S, Ann. Scuola Norm. Sup. Pisa, 23 (1969), 451-459. | Numdam | MR 41 #2053 | Zbl 0186.14003

[RaRuVe] J. P. Ramis, G. Ruget, J.-L. Verdier, Dualité relative en géométrie analytique complexe, Invent. Math., 13 (1971), 261-283. | MR 46 #7553 | Zbl 0218.14010

[Ru1] Y. Ruan, Topological sigma model and Donaldson type invariants in Gromov theory, Duke Math. Journ., 83 (1996), 461-500. | MR 97d:58042 | Zbl 0864.53032

[Ru2] Y. Ruan, Virtual neighborhoods and pseudo-holomorphic curves, preprint alg-geom/ 9611021. | Zbl 0967.53055

[RuTi1] Y. Ruan, G. Tian, A mathematical theory of quantum cohomology, Journ. Diff. Geom., 42 (1995), 259-367. | MR 96m:58033 | Zbl 0860.58005

[RuTi2] Y. Ruan, G. Tian, Higher genus symplectic invariants and sigma model coupled with gravity, Inv. Math., 130 (1997), 455-516. | MR 99d:58030 | Zbl 0904.58066

[Sa] I. Satake, The Gauss-Bonnet theorem for V-manifolds, Journ. Math. Soc. Japan, 9 (1957), 164-492. | MR 20 #2022 | Zbl 0080.37403

[Se] J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble, 6 (1956), 1-42. | Numdam | MR 18,511a | Zbl 0075.30401

[Si1] B. Siebert, Gromov-Witten invariants for general symplectic manifolds, preprint dg-ga/9608032, revised 12/97.

[Si2] B. Siebert, Global normal cones, virtual fundamental classes and Fulton's canonical classes, preprint 2/1997.

[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.