A note on invariant pseudoholomorphic curves (Q5890272)

The authors deal with a closed symplectic 4-manifold \((X,\omega)\). If a finite cyclic group \(G\) acts semifreely, holomorphically on \(X\), then there is a smooth structure on the quotient \(X'=X/G\) such that the projection \(\pi :X\to X'\) is a Lipschitz map. Let \(L\to X\) be the \(\text{Spin}^c\)-structure on \(X\) pulled back from a \(\text{Spin}^c\)-structure \(L'\to X'\) and \(b_2^+(X')\geq 2\).NEWLINENEWLINENEWLINEIn the paper under review, the equivariant version of the Taubes theorem is proved. More precisely, if the Seibert-Witten invariant \(\text{SW}(L')\neq 0\) and \(L=E\otimes K^{-1}\otimes E\), then there exists a \(G\)-invariant pseudoholomorphic curve \(u:C\to X\) such that \(u(C)\) represents the fundamental class of the Poincaré dual \(c_1(E)\).