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Math-2572 · mathematics · Algebraic geometry

Virasoro constraints for projective complete intersections

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every smooth complete intersection X in complex projective space, does its ordinary unreduced total descendant potential satisfy every Virasoro constraint in every genus and curve class, allowing arbitrary complex cohomology insertions?

Short English statement

For every smooth complete intersection X in complex projective space, does its ordinary unreduced total descendant potential satisfy every Virasoro constraint in every genus and curve class, allowing arbitrary complex cohomology insertions?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2572-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2572

Manuscript 1 (family 065, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Virasoro constraints for projective complete intersections

OpenAI's claim

Ordinary descendant Virasoro constraints are claimed for smooth projective complete intersections in every genus and curve class, retaining primitive and odd cohomology without semisimplicity. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $X\subset\mathbb P^N_{\mathbb C}$ be any smooth complete intersection. Let $F_g^X$ be its connected ordinary, unreduced genus-$g$ descendant potential, with cotangent classes at stable-map markings, arbitrary cohomology insertions and separate Novikov symbols for actual integral curve classes. Set $Z_X=\exp(\sum_{g\ge0}\hbar^{g-1}F_g^X)$. OpenAI claims the formal coefficientwise identities \[ L_k^X Z_X=0\qquad(k\ge-1). \] Here $L_k^X$ are the manuscript's normally ordered Virasoro operators built from the Poincaré pairing, first Hodge grading $\mu|_{H^{p,q}}=p-\dim(X)/2$ and $R=c_1(TX)\cup$, with its stated scalar normalization. The quantifiers include all genera, individual curve classes and primitive or odd insertions; these are descendant rather than ancestor constraints.

Outline of the argument

Manuscript 1 (family 065, 2026-09-24): Relative caps create and detect bounded seam-contact states with coefficientwise finite inversions. Pairing each primitive error tensor with its conjugate realizes a squared norm through these tests; Hodge--Riemann positivity then kills the full tensor without limiting the number of primitive insertions.

Formal verification

Family 065 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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