Hirzebruch's 1954 Problem List — Problem 7

Definitions and mathematical statement

Let $S^2=\{x\in{\mathbb{R}}^3:\|x\|=1\}$, and let $M=S^2\times S^2$. Under the usual interpretation that $M$ carries its standard product smooth structure, a complex structure on $M$ is an integrable smooth bundle endomorphism $J:TM\to TM$ satisfying $J^2=-\operatorname{id}_{TM}$. Integrability means that $J$ is induced by an atlas of complex coordinate charts with holomorphic transition maps; the resulting complex manifold has complex dimension two.

Classification problem. Determine all complex structures on $S^2\times S^2$.

Unresolved qualifications. The supplied statement does not specify an equivalence relation for this classification. If classification up to biholomorphism is intended, the target is \[ \left\{X:\ X\text{ is a compact connected complex surface and its underlying smooth manifold is diffeomorphic to }M\right\}/\cong_{\mathrm{bihol}}, \] where $X\cong_{\mathrm{bihol}}Y$ means that there is a holomorphic bijection $X\to Y$ with holomorphic inverse. Equivalently, complex structures $J_1,J_2$ on the fixed smooth manifold are identified when there is a diffeomorphism $f:M\to M$ such that ${\,\mathrm{d}} f\circ J_1=J_2\circ{\,\mathrm{d}} f$. This is a conditional interpretation, not an equivalence relation explicitly supplied by the source. The source also does not explicitly distinguish the standard smooth-product interpretation from a classification of complex surfaces merely homeomorphic to $M$, nor does it impose compatibility with a fixed orientation.

Short English statement

Determine all complex surface structures on $S^2\times S^2$. The intended equivalence relation and underlying smooth or topological interpretation require clarification. The supplied registry labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $800058$, source problem code catalogue entry, title "Hirzebruch's 1954 Problem List --- Problem 7." The record attributes the entry to Kotschick, with proposed year 2013, and labels it open.
• Kotschick, Updates on Hirzebruch's 1954 Problem List (2013), Problem 7. Supplied source URL: https://arxiv.org/abs/1305.4623. The record identifies the source list with Hirzebruch's 1954 problem list.
• Supplied provenance reports access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details, not a claim of independent source access or verification. The registry record was created and updated on 2026-10-16.