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Virasoro Constraints under Projectivization
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Abstract
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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Related prior literature mentioned by others
- Tom Coates, Alexander Givental and Hsian-Hua Tseng, Virasoro Constraints for Toric Bundles (Forum of Mathematics, Pi 12, e4, 2024; arXiv:1508.06282). The family exposition identifies this transfer theorem for toric bundles as an input to the complete-intersection induction and background for the proposed transfer to arbitrary projective bundles. [family-level]
Possibly relevant literature (found by the agent, not checked)
- H.-H. Tseng, A note on Virasoro constraints for products (arXiv:2603.22486, March 2026). Studies Virasoro constraints for a product variety when one factor has semisimple quantum cohomology. This is a trivial-fibration case of the base-to-total-space transfer that the paper's main theorem addresses.
- I. Ciocan-Fontanine, Y. Koto, A mirror theorem for partial flag bundles (arXiv:2602.09602, February 2026). A genus-zero mirror theorem for partial flag bundles of possibly non-split vector bundles. It extends the Iritani–Koto projective-bundle results that the paper uses in its comparison with the semisimple case.
- H. Fan, Y.-P. Lee, On Gromov–Witten theory of projective bundles (arXiv:1607.00740, 2016). Shows that the genus-zero equivariant GW theory of projective bundles over GKM bases depends only on equivariant Chern classes. This is a genus-zero companion to Fan's all-genus reconstruction and Chern-class dependence result, which the paper cites.
Digests and expositions
- AI Proved What?, Virasoro everywhere? Virasoro constraints for complete intersections and projective-bundle towers (episode 065, October 2026). This video and transcript summarize both claimed theorems and their degeneration, positivity and spectral-continuation arguments, with AI assistance and synthetic narration disclosed on the project’s About page. [AI-generated per source] [family-level]
Elsewhere
- Emergent Mind: family 065
Independent sites; their content is not reviewed here.
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}
}Family 065: 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.
Other papers in this family:
- Virasoro Constraints for Projective Complete Intersections · September 24, 2026