Finite group action and Gromov-Witten invariants in symplectic four-manifolds
This paper attempts to study the relationship between the relative Gromov-Witten moduli spaces (thus the corresponding invariants) of a symplectic \(4\)-manifold \(X\) and its quotient \(X/G\) by a finite group \(G\). The assumption is that the fixed point set \(F\) of the group action is the union of smooth \(S^2\)'s. The main result of the paper is Theorem \(4\) in section \(2\), in which the author shows that the moduli space of \(G\)-invariant genus \(0\) curves in \(X\) relative to \(F\) and that of genus \(0\) curves in \(X/G\) relative to the image of \(F\) are homeomorphic. Unfortunately, the proof only works in one direction, where a \(G\)-invariant curve of genus \(0\) on \(X\) induces a curve of the same genus on \(X/G\). In general, a genus \(0\) curve on \(X/G\) can be the quotient of a higher genus curve in \(X\). The more appropriate framework would be to study the orbifold Gromov-Witten theory on \([X/G]\).
scientific article; zbMATH DE number 5859369
scientific article