Symplectic surfaces in symplectic 4-manifolds (Q1811448)

Let \(X\) be a closed, minimal, symplectic 4-manifold with symplectic form \(\omega,\) and \(F\) a symplectic surface in \(X\) satisfying \(c_1(TX)[F] > 0.\) D. McDuff asked whether it follows that \(X\) must be rational or ruled. Partial results have been obtained by \textit{D. McDuff} in [J. Am. Math. Soc. 3, 679-712 (1990; Zbl 0723.53019); J. Am. Math. Soc. 5, 987-988 (1992; Zbl 0799.53039)]. In this paper the authors use McDuff's results and Seiberg-Witten and Gromov invariants to solve a restricted form of the problem, proving: Let \(X\) be a closed, connected, minimal symplectic 4-manifold containing a symplectic surface \(F\) with genus \(g(F) = g \geq 1\) and \(c_1(TX)[F] > g.\) Then \(X\) is rational or ruled.