Elliptic GW invariants of blowups along curves and surfaces (Q2569986)

The authors prove a result relating genus-one Gromov-Witten invariants of the blowup of a symplectic manifold along a curve or a surface to the invariants of the original manifold. The symplectic Gromov-Witten invariants \(\Psi^{M}_{A,1}(\alpha_1, \dots, \alpha_m)\) count \(J\)-holomorphic maps of curves of genus one into \(M\) with homology class \(A \in H_2(M)\) and \(m\) marked points intersecting cycles dual to the cohomology classes \(\alpha_1, \dots, \alpha_m\). This result extends work of the first author on the genus-zero case [Math. Z. 233, No. 4, 709--739 (2000; Zbl 0948.53046)]; [Compos. Math. 125, No. 3, 345--352 (2001; Zbl 1023.14029)]. The specific results proven are as follows: (1) Let \(M\) be a compact symplectic manifold, \(C\) a smooth curve in \(M\). If the genus of \(C\) is less than 2, assume that \(c_1(M)(C) \geq 0\). Denote by \(\pi: \tilde{M} \rightarrow M\) the blowup of \(M\) along \(C\). Assume that \(\pi^{!}(A)\) misses the exceptional locus of \(\tilde{M}\). Also assume that if \(\alpha_i\) has degree 1, it is supported away from \(C\). Then we have the equality of Gromov-Witten invariants \(\Psi^{M}_{(A,1)}(\alpha_1, \dots, \alpha_m)\) = \(\Psi^{\tilde{M}}_{(\pi^{!}(A),1)}(\pi^{*}(\alpha_1),\dots,\pi^{*}(\alpha_m))\). (2) Let \(M\) be a semipositive compact symplectic manifold, \(S\) a smooth surface in \(M\). Let \(\pi:\tilde{M} \rightarrow M\) denote the blowup along \(S\). Again assume that \(\pi^{!}(A)\) misses the exceptional locus and that if \(\alpha_i\) has degree at most 2, it is supported away from \(S\). Then the result of (1) also holds. The proofs of these results follow from application of the gluing formula for symplectic cuts due to \textit{A. Li} and \textit{Y. Ruan} [Invent. Math. 145, No. 1, 151--218 (2001; Zbl 1062.53073)] and \textit{E.-N. Ionel} and \textit{T. H. Parker} [Math. Res. Lett. 5, No. 5, 563--576 (1998; Zbl 0943.53046)]. In order to perform the symplectic blow-up, one removes a tubular neighborhood of the blowup locus, and reattaches it to the ambient manifold after collapsing the boundary by a Hamiltonian \(S^1\)-action. The authors then look carefully at the behavior of \(J\)-holomorphic maps with respect to the gluing operation.