×

zbMATH — the first resource for mathematics

Algebraic and symplectic Gromov-Witten invariants coincide. (English) Zbl 0970.14030
The purpose of this paper is to prove the following
Theorem: Let \(M\) be a complex projective manifold, \(R\in H_2(M,\mathbb Z)\), \(g,k\geq 0\). Let \({\mathcal C}_{R,g,k}(M)\) be the moduli space of stable maps of genus \(g\) with \(k\) marked points and representing class \(R\). Then the homology class associated to the algebraic virtual fundamental class (a Chow class on \({\mathcal C}_{R,g,k}(M))\) as in the paper by K. Behrend [Invent. Math. 127, 601-617 (1997; Zbl 0909.14007)] and the symplectic virtual fundamental class as in the paper by B. Siebert [“Gromov-Witten invariants for general symplectic manifolds”, preprint http://arXiv.org/abs/dg-ga/?9608005] coincide.

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14F35 Homotopy theory and fundamental groups in algebraic geometry
14D22 Fine and coarse moduli spaces
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
PDF BibTeX Cite
Full Text: DOI Numdam EuDML arXiv
References:
[1] K. BEHREND, GW-invariants in algebraic geometry, Inv. Math., 127 (1997), 601-617. · Zbl 0909.14007
[2] K. BEHREND, B. FANTECHI, The intrinsic normal cone, Inv. Math., 128 (1997), 45-88. · Zbl 0909.14006
[3] K. BEHREND, Y. MANIN, Stacks of stable maps and Gromov-Witten invariants, Duke. Math. Journ., 85 (1996), 1-60. · Zbl 0872.14019
[4] J. BINGENER, S. KOSAREW, Lokale modulräume in der analytischen geometrie I, II, Vieweg 1987. · Zbl 0644.32001
[5] A. DOUADY, Le problème des modules locaux pour LES espaces ℂ-analytiques compacts, Ann. Sci. École Norm. Sup. (4), 7 (1974), 569-602. · Zbl 0313.32036
[6] G. FISCHER, Complex analytic geometry, Lecture Notes Math., 538, Springer, 1976. · Zbl 0343.32002
[7] H. FLENNER, Über deformationen holomorpher abbildungen, Habilitationsschrift, Univ. Osnabrück, 1978.
[8] K. FUKAYA, K. ONO, Arnold conjecture and Gromov-Witten invariant, Warwick preprint 29/1996.
[9] W. FULTON, Intersection theory, Springer, 1984. · Zbl 0541.14005
[10] 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
[11] R. HARTSHORNE, Residues and duality, Lecture Notes Math., 20, Springer, 1966. · Zbl 0212.26101
[12] L. ILLUSIE, Complexe cotagent et déformations I, Lecture Notes Math., 239, Springer, 1971. · Zbl 0224.13014
[13] B. IVERSEN, Cohomology of sheaves, Springer, 1986. · Zbl 0559.55001
[14] T. KAWASAKI, The signature theorem for V-manifolds, Topology, 17 (1978), 75-83. · Zbl 0392.58009
[15] F. KNUDSON, The projectivity of the moduli space of stable curves, II: The stacks Mg, n, Math. Scand., 52 (1983), 161-199. · Zbl 0544.14020
[16] B. KAUP, L. KAUP, Holomorphic functions of several variables, de Gruyter, 1983. · Zbl 0528.32001
[17] J. LI, G. TIAN, Virtual moduli cycles and GW-invariants of algebraic varieties, Journal Amer. Math. Soc., 11 (1998), 119-174. · Zbl 0912.14004
[18] J. LI, G. TIAN, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, preprint alg-geom/9608032. · Zbl 0978.53136
[19] J. LI, G. TIAN, Comparison of the algebraic and the symplectic Gromov-Witten invariants, preprint alg-geom/9712035. · Zbl 0983.53061
[20] J. LIPMAN, Notes on derived categories and derived functors, preprint, available from http://www.math.purdue.edu/ lipman.
[21] G. POURCIN, Théorème de douady au-dessus de S, Ann. Scuola Norm. Sup. Pisa, 23 (1969), 451-459. · Zbl 0186.14003
[22] J. P. RAMIS, G. RUGET, J.-L. VERDIER, Dualité relative en géométrie analytique complexe, Invent. Math., 13 (1971), 261-283. · Zbl 0218.14010
[23] Y. RUAN, Topological sigma model and Donaldson type invariants in Gromov theory, Duke Math. Journ., 83 (1996), 461-500. · Zbl 0864.53032
[24] Y. RUAN, Virtual neighborhoods and pseudo-holomorphic curves, preprint alg-geom/ 9611021. · Zbl 0967.53055
[25] Y. RUAN, G. TIAN, A mathematical theory of quantum cohomology, Journ. Diff. Geom., 42 (1995), 259-367. · Zbl 0860.58005
[26] Y. RUAN, G. TIAN, Higher genus symplectic invariants and sigma model coupled with gravity, Inv. Math., 130 (1997), 455-516. · Zbl 0904.58066
[27] I. SATAKE, The Gauss-Bonnet theorem for V-manifolds, Journ. Math. Soc. Japan, 9 (1957), 164-492. · Zbl 0080.37403
[28] J.-P. SERRE, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble, 6 (1956), 1-42. · Zbl 0075.30401
[29] B. SIEBERT, Gromov-Witten invariants for general symplectic manifolds, preprint dg-ga/9608032, revised 12/97.
[30] B. SIEBERT, Global normal cones, virtual fundamental classes and Fulton’s canonical classes, preprint 2/1997. · Zbl 1083.14066
[31] 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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.