Kirby Problem 4.94

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A Horikawa surface is a smooth minimal complex projective surface $S$ of general type on the Noether line: \[ c_1(S)^2=2p_g(S)-4,\qquad p_g(S)\ge3, \] equivalently $5c_1(S)^2=c_2(S)-36$. Here $p_g(S)=\dim_{\mathbb C}H^0(S,K_S)$, $K_S$ is the canonical line bundle, and Chern numbers are evaluations of the indicated Chern classes on the complex-oriented fundamental class. General type means Kodaira dimension two, and minimal means no holomorphic sphere of self-intersection $-1$. Deformation equivalence means being connected by a chain of smooth proper holomorphic families over connected complex bases, with the given surfaces occurring as fibers.

Equip each such $S$ with its canonical symplectic structure $\omega_S$, unique up to symplectomorphism. Its cohomology class is $c_1(K_S)$; it is obtained by normalized pluricanonical projective forms and smoothing the canonical model, as in Catanese's construction. This normalization fixes the form class and is not a choice of an arbitrary Kähler form. [S3]

Statement recorded in the supplied source.

For every pair of homotopy-equivalent Horikawa surfaces $S,S'$ lying in different deformation classes, are their underlying smooth four-manifolds diffeomorphic? Are their canonical symplectic manifolds symplectomorphic, that is, does there exist a diffeomorphism $F:S\to S'$ with \[ F^*\omega_{S'}=\omega_S? \] Both questions range over the whole indicated class of pairs, rather than only a particular odd-parameter example. [S2]

Short English statement

Are homotopy-equivalent Horikawa surfaces in different complex deformation classes nevertheless diffeomorphic, and are their canonical symplectic structures symplectomorphic?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2970 (KP-4.94). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3—A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), author version, Problem 4.94 and Remarks, PDF page 267. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Fabrizio Catanese, Canonical symplectic structures and deformations of algebraic surfaces, arXiv:math/0608110v2 (2007), Theorem 1.3, page 3, and its proof. https://arxiv.org/pdf/math/0608110v2.