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geometry-topology / Complex geometry; six-manifolds; birational geometry

A complex structure on S2×S4S^{2}\times S^{4}

Determine whether the standard smooth manifold S2×S4S^{2}\times S^{4} admits a complex structure; the source constructs one conditionally on the cited compact complex threefold on S6S^{6}.

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55Significance / 100
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Acceptance criteria: Keep the construction conditional on the cited S6S^{6} source, separate complex structure from almost-complex structure, and do not claim independent verification or resolution of the cited S6S^{6} input.

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geometry-topologyAug 30, 2026Significance 55/100

Conditional complex structure on S2×S4S^{2}\times S^{4} reported

Prior state unknown→Conditional complex structure on S2×S4S^{2}\times S^{4} source-authenticated; cited S6S^{6} input and independent verification remain open

VibeMathed recorded: Conditional complex structure on S2×S4S^{2}\times S^{4} reported. Consult the linked registry entry for the original report and scope.

Source authenticatedA complex structure on S2×S4S^{2}\times S^{4} →
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Source authenticated

Assuming the cited compact complex threefold XsphX_{\mathrm{sph}} on the smooth S6S^{6} is valid, blowing up its fixed curve, quotienting by the order-two action, and applying an Atiyah flop produces a complex threefold X+X_{+} diffeomorphic to the standard smooth manifold S2×S4S^{2}\times S^{4}.

The construction takes the existence, topology, and group-action results for XsphX_{\mathrm{sph}} from the cited paper as input. MFN read the arXiv source but did not independently check those inputs, the Wall-classification calculations, or the flop construction.

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