Kirby Problem 4.92

Definitions and mathematical statement

Let $(X,\omega)$ be a closed connected symplectic four-manifold: $\omega$ is a smooth closed two-form nondegenerate on every tangent space. Choose an $\omega$-compatible almost-complex structure $J$, meaning $J^2=-1$ and $\omega(\cdot,J\cdot)$ is a positive-definite metric. The first Chern class $c_1(X,\omega)=c_1(TX,J)\in H^2(X;\mathbb Z)$ is independent of the compatible choice.

Call $(X,\omega)$ symplectic Calabi--Yau when this integral class is torsion: some positive integer multiple is zero. It need not vanish integrally. Must the underlying smooth manifold be diffeomorphic to one of the following: the standard K3 surface, the standard Enriques surface, or the total space of a smooth $T^2$ bundle over $T^2$?

A K3 surface may be modeled by a nonsingular quartic in $\mathbb{CP}^3$. An Enriques surface is the quotient of a complex K3 surface by a fixed-point-free holomorphic involution. A torus bundle is a smooth locally trivial projection onto $T^2=S^1\times S^1$ with fiber $T^2$; it is not required to be a product or principal bundle.

The proposed classification concerns diffeomorphism type, not symplectomorphism or deformation class of $\omega$. It allows different fundamental groups among the listed cases. Agreement merely of rational homology with one listed manifold is weaker than the requested conclusion. In particular, replacing torsion of $c_1$ by the numerical equality $c_1^2=0$ would substantially enlarge the hypothesis.

Short English statement

Are K3, Enriques, and torus-bundle smooth types the complete list of symplectic Calabi–Yau four-manifolds?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2968 (KP-4.92), 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 4.92. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.