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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.

MSC:

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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