A brief tour of GW invariants (Q2739545)

Let \((X,J,\omega)\) be a smooth symplectic manifold. For a non-negative integer \(k\) let \(N\) be a Riemann surface of genus \(g\) with \(k\) marked points \(x_1,\ldots ,x_k \in N\). For a homology class \(A\in H_2(X,Z)\) let \(\mathcal{M}_{g,k}(X,A)\) denote the moduli space of equivalence classes of \(J\)-holomorphic maps \(f\colon N \rightarrow X\) such that \(f_*([N]) = A\). Two \(J\)-holomorphic maps \(f\colon N \rightarrow X\) and \(f'\colon N' \rightarrow X'\) are said to be equivalent if there is an isomorphism \(\rho\colon N \rightarrow N'\) such that \(f'\circ \rho = f\). There is an evaluation map \(\text{ev}\colon \mathcal{M}_{g,k}(X,A) \rightarrow X^k\) defined by \(\text{ev}(f,x_1,\ldots , x_k)=(f(x_1), \ldots , f(x_k))\). Under certain conditions on \(\omega\) and \(J\) the preimage of a cycle \(\text{ev}^{-1}(\alpha) \subset \mathcal{M}_{g,k}(X,A)\) will be a finite oriented set for \([\alpha] \in H_*(X^k)\). The algebraic count of this oriented set is the GW-invariant of \((X,J,\omega)\). The GW-invariant of \((X,J,\omega)\) is equivalent to \(\int_{[\mathcal{M}_{g,k}(X,A)]}\text{ev}^*(\alpha)\), where \(\alpha \in H^*(X^k, Q)\) and \([\mathcal{M}_{g,k}(X,A)]\) is a homology class. The class \([\mathcal{M}_{g,k}(X,A)]\) is called a virtual moduli cycle. NEWLINENEWLINENEWLINEIn this paper, the authors present the construction of Gromov-Witten invariants and virtual moduli cycles. First, they discuss GW-invariants for semi-positive manifolds as defined by \textit{Y. Ruan} and \textit{G. Tian} [J. Differ. Geom. 42, 259-367 (1995; Zbl 0860.58005)], then they present the algebraic construction of virtual moduli cycles for smooth projective varieties and describe the analytic construction of virtual moduli cycles for general symplectic manifolds. Finally, the authors show that for a smooth projective variety with a Kähler form \(\omega\) the algebraically and analytically constructed GW-invariants coincide.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00007].