Result 065, Algebraic and complex geometry

Virasoro constraints for complete intersections and projective-bundle towers

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.

Proof

The bigger picture

Why it matters

Counting curves inside geometric spaces produces vast families of mathematical invariants. Virasoro constraints organize these numbers through an infinite system of identities; the reported results extend that structure to broad families of spaces and constructions built from them.

What changes?

The unreviewed manuscripts report full ordinary, unreduced descendant constraints for smooth complete intersections, spaces defined by polynomial equations of the expected codimension in complex projective space. Coverage includes every genus, each integral curve class and arbitrary cohomology insertions, including primitive and odd classes, without semisimplicity assumptions. Conditional on a smooth projective complex base satisfying the constraints, they pass to projectivizations of algebraic vector bundles of rank at least two, without splitting or positivity requirements, and to successive towers.

What does that help mathematicians do?

Descendant invariants incorporate extra geometric data at marked points on curves, while cohomology insertions specify conditions imposed there. The claimed identities therefore relate counts with many different kinds of conditions, not just a restricted selection. Researchers could use these relations to check computations and deduce dependencies among invariants throughout the stated families. Inclusion of primitive and odd classes means those relations also cover contributions that a narrower treatment could miss.

Are there practical applications?

The immediate value is foundational: a way to preserve a detailed structure in curve-counting theory while constructing new spaces. Projectivization replaces each vector-space fiber of a bundle by its space of lines. The reported inheritance result can be repeated, giving constraints for entire towers whenever the starting base satisfies them. It does not by itself provide a general algorithm for computing every invariant.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Virasoro Constraints under Projectivization

October 5, 2026 41 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 24, 2026 48 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.