Algebraic and complex geometry · Result 065

Virasoro constraints for complete intersections and projective-bundle towers

2 papers

Proves the full ordinary unreduced descendant Virasoro conjecture for smooth complete intersections in complex projective space, in every genus and curve class with arbitrary cohomology insertions. The constraints also pass from any smooth projective complex base satisfying them to the projectivization of every algebraic vector bundle of rank at least two, and hence to projective-bundle towers.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

Virasoro Constraints under Projectivization

OpenAI · October 5, 2026

We prove that full ordinary descendant Virasoro constraints pass from a smooth projective complex base to the projectivization of any algebraic vector bundle of rank at least two. The bundle need not split and satisfies no positivity requirement. Assuming the full constraints on the base, the conclusion includes every genus, each individual integral curve class, and all cohomology insertions, including primitive and odd classes. The result also applies successively to towers of projective bundles.

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Cite

@misc{OAI:Virasoro-Constraints-under-Projectivization-October-5-2026,
  author = {{OpenAI}},
  title = {{Virasoro Constraints under Projectivization}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026/virasoro-constraints-under-projectivization.pdf}{OAI:Virasoro-Constraints-under-Projectivization-October-5-2026}},
  year = {2026}
}

Virasoro Constraints for Projective Complete Intersections

OpenAI · September 24, 2026

We prove the Virasoro conjecture for the ordinary descendant Gromov–Witten theory of smooth complete intersections in projective space, in every genus and curve class and with arbitrary cohomology insertions. This includes primitive and odd cohomology classes, with no semisimplicity assumption.

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Cite

@misc{OAI:Virasoro-Constraints-for-Projective-Complete-Intersections-September-24-2026,
  author = {{OpenAI}},
  title = {{Virasoro Constraints for Projective Complete Intersections}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Virasoro-Constraints-for-Projective-Complete-Intersections-September-24-2026/article.pdf}{OAI:Virasoro-Constraints-for-Projective-Complete-Intersections-September-24-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.