View source for The Virasoro conjecture for complete intersections

The '''Virasoro conjecture''' in Gromov–Witten theory asserts that the total descendant potential of a smooth projective variety, the generating function of its curve counts with descendant insertions in all genera, is annihilated by a family of differential operators ''L''<sub>''k''</sub>, ''k'' ≥ −1, built from the Poincaré pairing, a Hodge grading and the first Chern class. Its simplest instance is the intersection theory of moduli spaces of curves, where Witten's conjecture and Kontsevich's theorem identify the generating function with a solution of the KdV hierarchy. Two manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), dated 24 September and 5 October 2026 and grouped as its result 065, claim the conjecture for every smooth complete intersection in complex projective space and its transfer to projective bundles. The family has no Lean formalization in the repository's catalogue. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.

== Background ==
* As recounted in the introductions of the manuscripts:
** Eguchi, Hori and Xiong proposed a target-dependent Virasoro system for Fano varieties, and Eguchi, Jinzenji and Xiong extended it to odd and off-diagonal Hodge classes, crediting Katz's proposal to use a Hodge index in the grading.
** The genus-zero constraints are known in general: Liu and Tian proved the decisive ''L''<sub>1</sub> and ''L''<sub>2</sub> identities, and Dubrovin and Zhang proved the genus-zero Frobenius-manifold formulation and the semisimple genus-one case.
** Among all-genus cases, Getzler proved the constraints for Calabi–Yau targets with vanishing first Chern class and first cohomology, Okounkov and Pandharipande proved the relative constraints for target curves, and Givental's quantization with Teleman's classification covers homogeneous semisimple theories. Guo, Zhang and Zhou proved genus-one constraints on the ambient state space of smooth Fano complete intersections.
** Primitive cohomology classes need not extend individually through a degeneration, which is the difficulty for complete intersections whose quantum cohomology is not semisimple.

== The original results ==
* '''Claim.''' Theorem 1.1 of ''Virasoro Constraints for Projective Complete Intersections'' states that the ordinary, unreduced total descendant potential of every smooth complete intersection in complex projective space satisfies the Virasoro constraints in every genus and curve class, with arbitrary insertions from its complex cohomology. The manuscript states that primitive and odd insertions are included, with no semisimplicity hypothesis.
* The companion ''Virasoro Constraints under Projectivization'' states, in its main theorem, that if the full constraints hold for a smooth connected projective complex variety ''B'', they hold for the projectivization of every algebraic vector bundle of rank at least two on ''B''. It states that the bundle need not split and is subject to no positivity condition, and that the result applies successively to towers of projective bundles.
* ''Method, as described:'' the complete-intersection manuscript factors one defining equation to obtain a degeneration of the variety into a complete intersection of smaller degree and the blowup of another along a center of smaller dimension, and argues by induction. The failure of each constraint is treated as an error tensor and tested by contractions on the first page of the weight spectral sequence; relative caps reproduce the contact pairing at each seam, an alternating sum over switches along chains of ruled components cancels, and the Hodge–Riemann form on primitive cohomology turns the pairing of an error with its conjugate into a squared norm, so the error vanishes. The projectivization manuscript embeds the bundle and the base in a larger projective bundle with a circle action, relates their theories by localization, separates Iritani's shift operator into spectral blocks and continues the equations from the branch belonging to the base to the branches belonging to the bundle.
* '''Sources''' (openai/math repository, commit <code>7ef5c0c</code>): [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/Virasoro-Constraints-for-Projective-Complete-Intersections-September-24-2026 Virasoro Constraints for Projective Complete Intersections], dated 24 September 2026; [https://github.com/openai/math/tree/7ef5c0cc5ca4b0e072b7a050fca09ab24bc6fa3a/preprints/Virasoro-Constraints-under-Projectivization-October-5-2026 Virasoro Constraints under Projectivization], dated 5 October 2026.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 30 October 2026)
|-
| 2026-10-12 || ''AI Proved What?'', episode 065 [https://aiprovedwhat.com/episodes/065-virasoro-constraints-complete-intersections-projective/ Virasoro everywhere?], remade version of 26 October 2026 || Reading (exposition) || Video explainer of both claims and their history. The episode states that the results concern the ordinary unreduced descendant theory of complete intersections in projective space and of projective bundles over bases where the constraints are already known, and do not prove the conjecture for every smooth projective variety. It lists the inputs as degeneration formulas and the reconstruction of Argüz, Bousseau, Pandharipande and Zvonkine, Steenbrink's limiting Hodge structures, the toric-bundle transfer of Coates, Givental and Tseng, Iritani's shift operators and quantum Riemann–Roch. || Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as preprints: OpenAI, dated September 24 and October 5, 2026, released October 6; under expert review: not yet; peer reviewed: not yet; checked in Lean: no Lean proof published. The page states that it found no public assessment by specialists.
|-
| 2026-10-16
| J. Jardine, Lattice24, referee reviews [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_199.json rank_199] (complete intersections) and [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_301.json rank_301] (projectivization) in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Both reviews report no gap.
* The complete-intersection review has three minor notes: an extended product rule for groups of possibly disconnected components, on which the cancellation depends, is given only as a sketch; the passage to the limiting mixed Hodge structure needs monodromy compatibility that holds but is not stated; and the use of the toric-bundle transfer for bases with odd and off-diagonal cohomology is argued only qualitatively.
* The projectivization review has two minor notes: the extension of ancestor localization to families of orbit lines is argued in outline, and the fixed-point formula for the shift operator is imported from Iritani with a brief check that the reviewer's own consistency checks agree with.
| Part of an audit that its README describes as complete for the collection's manuscripts without a Lean formalization. Each review states which sections were read in full. The reviews are published under CC BY 4.0.
|}

== Notes ==
* The background summarizes the manuscripts' own account of earlier work.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]