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

TL;DR

The paper develops a symplectic analogue of relative affine-ruledness by studying pairs (X,ω,D) with negative adjoint pairing $\omega\cdot(K_{\omega}+[D])<0$, establishing that in this regime the divisor complement X∖D is symplectically affine-ruled and, in the rational case, is governed by unicuspidal rational curves of index one whose normal crossing resolutions yield zero-self-intersection fibers. The authors build a comprehensive toolkit—divisor operations, Seiberg–Witten/Gromov invariants, divisor-adapted almost complex structures, and curve cone results—to reduce complex divisor configurations to quasi-minimal models and to realize the desired foliations via $J$-holomorphic curves, using MO’s embedded-curve criterion and Zhang’s curve cone theorem. A key outcome is that such pairs are symplectic deformation equivalent to Kähler pairs, and the open dense subset of the divisor complement inherits a product-like, deformation-equivalent symplectic structure. The work unifies relative symplectic geometry with algebraic analogues, and provides concrete mechanisms (unicuspidal curves, normal crossing resolutions, and inflation) to pass from a negative adjoint condition to a geometric foliation and to deformation stability, with implications for symplectic log Calabi–Yau structures and deformation theory.

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.

Figures (12)

• Figure 1: After the normal crossing resolution, $\tilde{C}$ will be an embedded sphere of self-intersection $0$.
• Figure 2: Four types of blowups, where the dashed curve means it is not contained in $\tilde{D}$.
• Figure 3: The normal crossing resolution for a $(5,2)$-cusp. The intersection pattern in the last configuration implies that $[\tilde{C}]=[C]-2E_1-2E_2-E_3-E_4$.
• Figure 4: A comb-like configuration, where the dashed curves are not included in $D$.
• Figure 5: The curve configurations for quasi-minimal pairs of first/second kind, where the dashed curves are not included as part of $D$.

Theorems & Definitions (119)

• Theorem 1.1: LiuAiKoOhtaOnoc_1positive
• Theorem 1.3: FujitaMiysugiekodinfRusselaffineruledMiyanishi82affineruled
• Definition 1.4
• Definition 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8: =Theorem \ref{['thm:deformation']}
• Corollary 1.9
• proof
• Proposition 1.10