Symplectic log Kodaira dimension $-\infty$, affine-ruledness and unicuspidal rational curves

Abstract

Given a closed symplectic $4$-manifold $(X,ω)$, a collection $D$ of embedded symplectic submanifolds satisfying certain normal crossing conditions is called a symplectic divisor. In this paper, we consider the pair $(X,ω,D)$ with symplectic log Kodaira dimension $-\infty$ in the spirit of Li-Zhang. We introduce the notion of symplectic affine-ruledness, which characterizes the divisor complement $X\setminus D$ as being foliated by symplectic punctured spheres. We establish a symplectic analogue of a theorem by Fujita-Miyanishi-Sugie-Russell in the algebraic settings which describes smooth open algebraic surfaces with $\overlineκ=-\infty$ as containing a Zariski open subset isomorphic to the product between a curve and the affine line. When $X$ is a rational manifold, the foliation is given by certain unicuspidal rational curves of index one with cusp singularities located at the intersection point in $D$. We utilize the correspondence between such singular curves and embedded curves in its normal crossing resolution recently highlighted by McDuff-Siegel, and also a criterion for the existence of embedded curves in the relative settings by McDuff-Opshtein. Another main technical input is Zhang's curve cone theorem for tamed almost complex $4$-manifolds, which is crucial in reducing the complexity of divisors. We also investigate the symplectic deformation properties of divisors and show that such pairs are deformation equivalent to Kähler pairs. As a corollary, the restriction of the symplectic structure $ω$ on an open dense subset in the divisor complement $X\setminus D$ is deformation equivalent to the standard product symplectic structure.