Bounded Negativity and Symplectic 4-Manifolds

TL;DR

This paper proves that in certain symplectic 4-manifolds with negative Kodaira dimension, the self-intersection number of symplectic curves is bounded below, supporting the bounded negativity conjecture for specific complex surfaces.

Contribution

It establishes a lower bound on the self-intersection of symplectic curves in symplectic 4-manifolds with negative Kodaira dimension, linking symplectic and complex geometry.

Findings

Bound on self-intersection numbers in symplectic 4-manifolds

Implication for the bounded negativity conjecture in algebraic geometry

Discussion of related negativity problems in different structures

Abstract

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 $\kappa=-\infty$.