Family Gromov-Witten invariants for Kähler surfaces. (English) Zbl 1059.53070

Gromov-Witten invariants are numbers of holomorphic curves in a symplectic manifold \(M\). To define them using the analytic approach, one chooses an almost complex structure \(J\) compatible with the symplectic structure and considers the set of maps \(f: S\to M\) from a Riemann surface \(S\) which satisfy the following \(J\)-holomorphic map equation: \(\overline\partial_ Jf=0\). After compactifying the moduli space of such maps, one imposes constraints requiring, for example, that the image of the map passes through specified points. With the right number of constraints and a generic \(J\), the number of such maps is finite. That number is a Gromov-Witten invariant and depends only on the symplectic structure of \(M\).
In this paper, the author uses analytic methods to define family Gromov-Witten invariants for Kähler surfaces. It is proven that they are well-defined invariants of the deformation class of the Kähler structure.


53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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[1] W. Barth, C. Peters, and A. Van de Ven, Compact Complex Surfaces , Ergeb. Math. Grenzgeb. (3) 4 , Springer, Berlin, 1984. · Zbl 0718.14023
[2] K. Behrend and B. Fantechi, in preparation.
[3] A. L. Besse, Einstein Manifolds , Ergeb. Math. Grenzgeb. (3) 10 , Springer, Berlin, 1987. · Zbl 0613.53001
[4] J. Bryan and N. C. Leung, “Counting curves on irrational surfaces” in Surveys in Differential Geometry: Differential Geometry Inspired by String Theory , Surv. Differ. Geom. 5 , Int. Press, Boston, 1999, 313–339. · Zbl 0999.14019
[5] –. –. –. –., Genenerating functions for the number of curves on abelian surfaces , Duke Math. J. 99 (1999), 311–328. · Zbl 0976.14033
[6] –. –. –. –., The enumerative geometry of \(K3\) surfaces and modular forms , J. Amer. Math. Soc. 13 (2000), 371–410. JSTOR: · Zbl 0963.14031
[7] S. K. Donaldson, “Yang-Mills invariants of four-manifolds” in Geometry of Low-Dimensional Manifolds, 1: Gauge Theory and Algebraic Surfaces (Durham, U.K., 1989) , ed. S. K. Donaldson and C. B. Thomas, London Math. Soc. Lecture Note Ser. 150 , Cambrige Univ. Press, Cambridge, 1990, 5–40. · Zbl 0836.57012
[8] R. Friedman and J. W. Morgan, Smooth Four-Manifolds and Complex Surfaces , Ergeb. Math. Grenzgeb. (3) 27 , Springer, Berlin, 1994. · Zbl 0817.14017
[9] L. Göttsche, A conjectural generating function for numbers of curves on surfaces , Comm. Math. Phys. 196 (1998), 523–533. · Zbl 0934.14038
[10] P. Griffiths and J. Harris, Principles of Algebraic Geometry , Pure Appl. Math., Wiley, New York, 1978. · Zbl 0408.14001
[11] E.-N. Ionel and T. H. Parker, Relative Gromov-Witten Invariants , Ann. of Math. (2) 157 (2003), 45–96. JSTOR: · Zbl 1039.53101
[12] ——–, The symplectic sum formula for Gromov-Witten invariants , to appear in Ann. of Math. (2), preprint. JSTOR: links.jstor.org
[13] S. Ivashkovich and V. Shevchishin, Gromov compactness theorem for \(J\)-complex curves with boundary , Internat. Math. Res. Notices 2000 , no. 22, 1167–1206. · Zbl 0994.53010
[14] S. Kleiman and R. Piene, “Enumerating singular curves on surfaces” in Algebraic Geometry: Hirzebruch 70 (Warsaw, 1998) , Contemp. Math. 241 , Amer. Math. Soc., Providence, 1999, 209–238. · Zbl 0953.14031
[15] P. B. Kronheimer and T. S. Mrowka, Embedded surfaces and the structure of Donaldson’s polynomial invariants , J. Differential Geom. 41 (1995), 573–734. · Zbl 0842.57022
[16] J. Li and G. Tian, “Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds” in Topics in Symplectic \(4\)-Manifolds (Irvine, Calif., 1996) , First Int. Press Lect. Ser. 1 , Internat. Press, Cambridge, Mass., 1998, 47–83. · Zbl 0978.53136
[17] T. H. Parker, “Compactified moduli spaces of pseudo-holomorphic curves” in Mirror Symmetry, III (Montreal, 1995) , AMS/IP Stud. Adv. Math. 10 , Amer. Math. Soc., Providence, 1999, 77–113. · Zbl 0927.58003
[18] T. H. Parker and J. Wolfson, Pseudo-holomorphic maps and bubble trees , J. Geom. Anal. 3 (1993), 63–98. · Zbl 0759.53023
[19] I. Vainsencher, Enumeration of n-fold tangent hyperplanes to a surface , J. Algebraic Geom. 4 (1995), 503–526. · Zbl 0928.14035
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