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Calabi’s problem for genus one

Does every compact complex structure on the smooth manifold T2×S4T^2\times S^4 split biholomorphically as a product of complex manifolds? Equivalently, for every integrable complex structure JJ on T2×S4T^2\times S^4, is (T2×S4,J)(T^2\times S^4,J) biholomorphic to a product complex manifold?

References

Primary source

arXiv

Additional references

complex-geometrytopology
Posted · edited

Progress summary

Refreshed
Claimed solved

An October 2026 unrefereed preprint claims to settle the genus-one case by constructing a compact complex structure that does not split as a product.

The genus-one case concerns whether a compact complex structure on T2×S4T^2 \times S^4 must split biholomorphically as a product. Min Gao, Zizhou Tang, and Wenjiao Yan claim a negative answer.

October 2026 construction

The preprint constructs a compact non-Kähler complex threefold on T2×S4T^2 \times S^4 and proves, as claimed, that it is not biholomorphic to a product. This settles the specifically tracked genus-one case if correct, but the preprint is unrefereed and does not address the broader problem.

Current status (as of October 2026): The genus-one case is claimed solved by an unrefereed preprint, while the broader Calabi problem remains open.

Sources

Solutions 0

No solutions have been posted yet.