Bounded Negativity and Symplectic 4-Manifolds

Let $(M,ω)$ be a symplectic 4-manifold of negative Kodaira dimension. Let $C$ be an $ω$-symplectic curve, $J$-holomorphic for some $J$ tamed by $ω$. Then $[C]^2$ is bounded below by a constant depending only on $ω$. 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 $κ=-\infty$.

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