Counting curves on elliptic ruled surface
The authors calculate the symplectic Gromov-Witten invariants of \(S^2\times T^2\) by using Kähler structures [cf. \textit{M. Gromov}, Invent. Math. 82, 307-347 (1985; Zbl 0592.53025)]. In alternative terms, the results concern the number of embedded genus \(g\) curves in the homology class \[ A_{\ell, d}=l[S^2]+ d[T^2]. \] The first result determines the number \(N_1(A_{\ell,1})=2\) of embedded genus 1 curves in \(A_{\ell,1}\) passing through \(2\ell\) points. The second result determines the generating function (not stated here) for the number \(N_g(A_{1,d})\) (or \(n_g(A_{1,d})\), resp.) of genus \(g\) curves in class \(A_{1,d}\) passing through \(g+1\) points (passing through \(g\) points and intersecting two circles which generate the first integral homology, respectively).
scientific article; zbMATH DE number 1879097
scientific article