Kirby Problem 4.96

Definitions and mathematical statement

A closed symplectic $4$-manifold $(X,\omega)$ is a compact boundaryless smooth $4$-manifold with a closed nondegenerate real $2$-form. An $\omega$-compatible almost complex structure is an endomorphism $J:TX\to TX$ with $J^2=-1$, preserving $\omega$ and satisfying $\omega(v,Jv)>0$ for nonzero $v$. The first Chern class $c_1(X,\omega)\in H^2(X;\mathbb Z)$ is the first Chern class of the complex tangent bundle $(TX,J)$; it is independent of the compatible choice of $J$. Write $[\omega]\in H^2(X;\mathbb R)$ for the de Rham class.

Do there exist two closed symplectic $4$-manifolds $(X_1,\omega_1),(X_2,\omega_2)$ and a diffeomorphism $f:X_1\to X_2$ such that \[ f^*c_1(X_2,\omega_2)=c_1(X_1,\omega_1),\qquad f^*[\omega_2]=[\omega_1], \] but no symplectomorphism between the two manifolds exists? A symplectomorphism would be some diffeomorphism $g:X_1\to X_2$ with $g^*\omega_2=\omega_1$ as differential forms.

The map excluded in the conclusion is every possible $g$, not just the particular $f$ witnessing the topological and cohomological agreements. Thus showing $f^*\omega_2\ne\omega_1$ is insufficient. Equivalently, after transporting forms along $f$, the problem seeks cohomologous symplectic forms on one smooth manifold with the same first Chern class but different symplectomorphism types. No deformation or path connecting those forms is assumed. All the displayed equalities are required for the same witnessing diffeomorphism, and the nonexistence conclusion allows every possible alternative diffeomorphism.

Short English statement

Can two closed symplectic four-manifolds have matching smooth type, symplectic cohomology class and first Chern class, yet fail to be symplectomorphic?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2972 (KP-4.96), 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 4.96. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.