Symplectic surfaces and generic $J$-holomorphic structures on 4-manifolds

Abstract

It is a well known fact that every embedded symplectic surface $\Sigma$ in a symplectic four-manifold $(X^4,\omega )$ can be made $J$-holomorphic for some almost-complex structure $J$ compatible with $\omega$. In this paper we investigate when such a structure $J$ can be chosen generically in the sense of Taubes. The main result is stated in Theorem 1.2. As an application of this result we give examples of smooth and non-empty Seiberg-Witten and Gromov-Witten moduli spaces whose associated invariants are zero.