Bounded Negativity and Symplectic 4-Manifolds

Let $(M,\omega )$ be a symplectic 4-manifold of negative Kodaira dimension. Let $C$ be an $\omega$-symplectic curve, $J$-holomorphic for some $J$ tamed by $\omega$. Then $[C{]}^2$ is bounded below by a constant depending only on $\omega$. Related bounded negativity problems for other structures are also briefly discussed. In particular, the symplectic result implies the bounded negativity conjecture for complex projective surfaces with Kodaira dimension $\kappa =-\mathrm{\infty }$.