Kirby Problem 4.91

Definitions and mathematical statement

A symplectic fake projective plane is a closed connected smooth four-manifold $X$ equipped with a closed nondegenerate two-form $\omega$, whose rational cohomology ring is isomorphic to that of $\mathbb{CP}^2$, but whose smooth manifold is not diffeomorphic to $\mathbb{CP}^2$. Use the symplectic orientation. Equivalently, its rational cohomology is generated by a degree-two class with nonzero square, with Betti numbers $b_1=b_3=0$ and $b_2=1$.

Give constructions of such symplectic manifolds by explicit topological or symplectic operations, with enough control to verify both the cohomology condition and the resulting smooth type. The construction request includes obtaining known examples by these methods; it is not itself a uniquely specified invariant to calculate.

The accompanying precise existence question asks whether a symplectic fake projective plane can fail to be a complex ball quotient. A complex ball quotient here is $\mathbb B^2/\Gamma$, where $\mathbb B^2\subset\mathbb C^2$ is the complex unit ball and $\Gamma\le\operatorname{PU}(2,1)$ is a torsion-free discrete cocompact subgroup acting by holomorphic isometries. Seek an $X$ not diffeomorphic to any such quotient.

Rational cohomology is the requirement: the fundamental group need not be trivial, and integral homology may contain torsion. A merely smooth fake projective plane without a symplectic form does not qualify. The classification is of underlying smooth manifolds admitting the required structures, rather than of one selected compatible complex structure.

Short English statement

Construct symplectic fake projective planes topologically, and determine whether one can lie outside the complex-ball-quotient class.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2967 (KP-4.91), 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 4.91. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.