Kirby Problem 4.100

Definitions and mathematical statement

Let $(X,\omega)$ be a closed connected symplectic four-manifold. The class $c_1(X,\omega)$ is the first Chern class of its tangent bundle with any compatible almost-complex structure; the resulting class does not depend on that choice. For $c\in H_2(X;\mathbb Z)$, denote evaluation of a cohomology class $a$ on $c$ by $\langle a,c\rangle$ and its self-intersection by $c\cdot c$.

Suppose $c$ has a connected oriented smoothly embedded representative $S$ satisfying \[ \langle[\omega],c\rangle>0, \qquad \langle c_1(X,\omega),c\rangle =\chi(S)+c\cdot c. \] Must $c$ have a connected embedded symplectic representative? Such a representative is an embedded surface $S'$ with $[S']=c$ for which $\omega|_{S'}$ is a positive area form, using the indicated orientation.

The conditions are positive symplectic area and equality in the symplectic adjunction formula for the genus of the given smooth representative. They are necessary for $S$ itself to be symplectic; the question asks whether they suffice for some representative of its class. The desired $S'$ need not be isotopic to $S$.

Its genus would necessarily equal that of $S$, since the adjunction formula and the two fixed pairings determine $\chi(S')$. No assumption of positive self-intersection, primitivity of $c$, or integrality of $[\omega]$ is made. Replacing the fixed symplectic form by a different one would change the problem.

Short English statement

Do positive symplectic area and the adjunction equality guarantee a symplectic representative of a homology class?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2976 (KP-4.100), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.100. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.