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Virasoro Constraints for Projective Complete Intersections
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Abstract
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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Related prior literature mentioned by others
- Hülya Argüz, Pierrick Bousseau, Rahul Pandharipande and Dimitri Zvonkine, Gromov–Witten Theory of Complete Intersections via Nodal Invariants (arXiv:2109.13323, 2021; revised 2023). The family exposition identifies this all-genus reconstruction using nodal invariants as the geometric basis for the manuscript’s induction on dimension and defining degrees.
- 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]
- C. Hertling, Yu. Manin and C. Teleman, An update on semisimple quantum cohomology and F-manifolds (arXiv:0803.2769, 19 March 2008). The family exposition cites its restriction of generically semisimple even quantum cohomology to Hodge–Tate targets without odd cohomology to explain the significance of the manuscript’s primitive and odd insertions.
Possibly relevant literature (found by the agent, not checked)
- D. Maulik, D. Ranganathan, Gromov–Witten theory, degenerations, and the tautological ring (arXiv:2510.04779, October 2025). Develops logarithmic degeneration and reconstruction techniques, including for complete intersections in products of projective spaces; this may bear on the paper's induction through degenerations $X\rightsquigarrow U\cup_D\widetilde M$.
- X. Hu, Big quantum cohomology of Fano complete intersections (arXiv:1501.03683, 2015, revised 2025). Studies genus-0 invariants of complete intersections with primitive insertions via monodromy and invariant theory; this may be relevant to the paper's treatment of primitive insertions, which it says "need not extend individually through a degeneration".
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
- VibeMathed: Problem record
- Emergent Mind: family 065
Independent sites; their content is not reviewed here.
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}
}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 under Projectivization · October 5, 2026