Kirby Problem 4.109

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $(X,\omega)$ be a closed symplectic four-manifold, meaning that $\omega$ is a closed nondegenerate real two-form. Let $\Sigma\subset X$ be a closed embedded symplectic surface: the restriction of $\omega$ to each oriented tangent plane of $\Sigma$ is positive. Suppose that, in real homology, \[ [\Sigma]=\operatorname{PD}(k[\omega])\in H_2(X;\mathbb R) \] for an integer $k$, where $\operatorname{PD}$ is Poincaré duality and the integral fundamental class is viewed in real homology. A Weinstein domain is a compact manifold $F$ with boundary, a one-form $\lambda$ such that $d\lambda$ is symplectic and its Liouville vector field $V$ points outward along the boundary, and a Morse function for which $V$ is gradient-like. Here $\iota_Vd\lambda=\lambda$; gradient-like means that the function increases strictly along $V$ away from its critical points, with the usual local gradient behavior there. In the complement convention of Giroux, compatibility with the given form means that there is a diffeomorphism \[ q:\operatorname{int}(F)\longrightarrow X\setminus\Sigma, \qquad q^*(\omega|_{X\setminus\Sigma})=d\lambda|_{\operatorname{int}(F)}. \] The symplectic complement is the interior of a Weinstein domain, rather than a completion obtained by attaching an infinite-volume end. [S3]

Statement recorded in the supplied source.

For every such symplectic surface $\Sigma$, does its complement $X\setminus\Sigma$ admit a Weinstein structure compatible with the restriction of the given form $\omega$? The question concerns each given representative of the indicated Poincaré-dual class, rather than the existence of some suitably chosen representative. [S2]

Short English statement

Must the complement of every symplectic surface in an integral multiple of the symplectic class carry a Weinstein structure whose symplectic form is the original restricted form?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2985 (KP-4.109). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3—A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), author version, Problem 4.109 and Remarks, PDF page 281. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Emmanuel Giroux, Remarks on Donaldson's symplectic submanifolds, arXiv:1803.05929v1 (2018), pages 2--3, Weinstein-domain definition and Theorem 2. https://arxiv.org/pdf/1803.05929v1.