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The Virasoro conjecture for smooth complete intersections in projective space (and its transfer to projective bundles)

For a smooth projective variety XX, let ZX=exp⁡(∑gℏg−1FgX)Z_X=\exp(\sum_g\hbar^{g-1}F_g^X) be the total descendant Gromov-Witten potential. Eguchi, Hori and Xiong (1997) proposed target-dependent differential operators LkXL_k^X, k≥−1k\ge-1, built from the Poincare pairing, a Hodge grading and cup product with c1(TX)c_1(TX), generalising the Witten-Kontsevich KdV constraints for a point; Eguchi, Jinzenji and Xiong extended them to odd and off-diagonal classes. The Virasoro conjecture asserts LkXZX=0L_k^XZ_X=0 for all k≥−1k\ge-1. Genus zero, Calabi-Yau targets with H1=0H^1=0, curves, toric bundles and semisimple theories (Givental-Teleman) were known, but non-semisimple higher-genus targets largely were not. Does the Virasoro conjecture hold, in every genus and curve class with arbitrary insertions, for every smooth complete intersection in projective space?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Algebraic geometry; Gromov-Witten theory
Posed by
Tohru Eguchi, Kentaro Hori and Chuan-Sheng Xiong (extended by Eguchi, Jinzenji and Xiong)
Year posed
1997
Years open
29y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
40 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Principal theorem: for a smooth complete intersection X⊂PCNX\subset\mathbb P^N_{\mathbb C}, LkXZX=0L_k^XZ_X=0 for all k≥−1k\ge-1 in Getzler's first-Hodge-degree superspace convention, coefficientwise in every genus and curve class, with arbitrary insertions from H∗(X;C)H^*(X;\mathbb C). Companion: if the full constraints hold for a smooth connected projective base BB, they hold for PB(E)\mathbb P_B(E) for any algebraic vector bundle EE of rank at least 2 (no splitting or positivity), hence for projective-bundle towers. The conjecture for general smooth projective varieties (for example general hypersurfaces in other ambients, or varieties with no degeneration to known pieces) remains open.

What the AI did

The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts: the complete-intersection theorem (September 24, 2026) and a projectivization transfer theorem (October 5, 2026).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript (Virasoro constraints for the ordinary unreduced descendant potential of every smooth complete intersection in PCN\mathbb P^N_{\mathbb C}, every genus and curve class, all insertions including primitive and odd classes, no semisimplicity) and Theorem 1.1 of the companion (V(B)⇒V(PB(E))\mathcal V(B)\Rightarrow\mathcal V(\mathbb P_B(E)) for every algebraic bundle of rank at least 2) were read against the conjecture as posed. The proof (degeneration with a weight spectral sequence, cap tests, a Hodge-Riemann positivity detector) was not refereed and there is no Lean formalization. The manuscripts use established degeneration and toric-bundle transfer results as stated inputs.

Sources

Changelog1 change

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