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The virtual moduli cycle. (English) Zbl 0958.53060

Eliashberg, Ya. (ed.) et al., Northern California symplectic geometry seminar. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 196(45), 73-102 (1999).
Let \((X,\omega)\) be a symplectic manifold with compatible almost complex structure \(J\). If \(J\) is generic then the space \({\mathcal M}_{0,k}(X, J,A)\) of unparametrized \(J\)-holomorphic spheres in class \(A\) with \(k\) marked points is a manifold which is usually not compact. But in many cases there is an evaluation map \(ev:{\mathcal M}_{0,k}(X, J,A)\to X^k\) representing a well defined element in homology \(H_*(X^k)\), which can be used to define the Gromov-Witten invariants. The author constructs the compactification of \({\mathcal M}_{0,k}(X, J,A)\) and shows how to use the compactification process of glueing in the domain, through an orbifold structure of the ambient domain, to endow the space with topology. Mainly the construction of the virtual moduli cycle which is used as a tool in the above mentioned construction of Gromov-Witten invariants is described. Using the Liu and Tian theory [Gang Liu and Gang Tian, J. Differ. Geom. 49, 1-74 (1998; Zbl 0917.58009)], which represents the virtual moduli cycle by the zero set of a suitable multi-section of a multi-bundle, the author shows how to assemble this zero set into an object which is called a branched pseudo-manifold and is constructed from a Fredholm section of a multi-bundle over a multi-fold.
For the entire collection see [Zbl 0930.00050].

MSC:

53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds

Citations:

Zbl 0917.58009
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