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Virtual neighborhoods and pseudo-holomorphic curves. (English) Zbl 0967.53055

The virtual neighborhood technique developed by the author in [Ronald J. Stern (ed.), Topics in symplectic \(4\)-manifolds, First Int. Press Lect. Ser. 1, 101-116 (1998; Zbl 0939.57024)] is extended to produce a stable compactification of the moduli space of pseudo-holomorphic curves in a compact symplectic manifold \((V,\omega)\). Here it is important to notice that the author does not need any semipositivity assumption on \([\omega]\) and \(c_1(V)\).
The author gives several applications of this method. An equivariant quantum cohomology is constructed using a generalisation of Gromov-Witten invariants to families of symplectic manifolds. It is shown that Floer homology can be defined on all compact symplectic manifolds. Finally, a version of the Arnol’d conjecture is proved for general compact symplectic manifolds \((V,\omega)\), which says that if a Hamiltonian symplectomorphism of \(V\) has only non-degenerate fixed points, then the number of fixed points is at least the total Betti number of \(V\).
Related results have been obtained by K. Fukaya and K. Ono [Topology 38, No. 5, 933-1048 (1999; Zbl 0946.53047)] and by G. Liu and T. Tian [J. Differ. Geom. 49, No. 1, 1-74 (1998; Zbl 0917.58009)].

MSC:

53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
58D27 Moduli problems for differential geometric structures
53D40 Symplectic aspects of Floer homology and cohomology
32Q65 Pseudoholomorphic curves
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