Kirby Problem 4.101

Definitions and mathematical statement

Let $\mathbb{CP}^2$ have its standard complex structure and Fubini--Study symplectic form $\omega_{FS}$, normalized so a projective line has area one. A smooth symplectic surface is a closed smoothly embedded surface $S$ on which $\omega_{FS}$ restricts to a positive area form, giving its orientation. Write $h\in H_2(\mathbb{CP}^2;\mathbb Z)$ for the class of a projective line. The degree of $S$ is the integer $d$ with $[S]=dh$; its symplectic area implies $d>0$.

A smooth complex plane curve of degree $d$ is the zero set of a homogeneous complex polynomial of degree $d$ in three variables, with nonsingular zero locus in projective space. Such a curve is a symplectic surface in the same homology class. A symplectic isotopy is a smooth family of embeddings of a fixed surface whose images remain symplectic at every time.

Is every smooth symplectic surface in $\mathbb{CP}^2$ symplectically isotopic to a smooth complex plane curve? Equivalently, for every $d\ge1$, is there exactly one symplectic isotopy class of embedded symplectic surfaces of degree $d$? The equivalence uses connectedness of the family of nonsingular complex curves of a fixed degree.

The ambient form remains $\omega_{FS}$ during the isotopy. A deformation passing through singular surfaces, an arbitrary smooth isotopy whose intermediate surfaces are not symplectic, or a change of ambient symplectic form would not meet the requested condition.

Short English statement

Is every embedded symplectic surface in the complex projective plane symplectically isotopic to a nonsingular algebraic curve of the same degree?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2977 (KP-4.101), 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 4.101. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.