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Math-2596 · mathematics · Algebraic geometry

Virasoro propagation through projective bundles

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For any smooth connected projective complex variety B and algebraic vector bundle E of rank at least two, if all ordinary descendant Virasoro constraints hold for B in every genus, curve class and cohomology insertion, must they also hold for the projective bundle of lines $P_B(E)$?

Short English statement

For any smooth connected projective complex variety B and algebraic vector bundle E of rank at least two, if all ordinary descendant Virasoro constraints hold for B in every genus, curve class and cohomology insertion, must they also hold for the projective bundle of lines $P_B(E)$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2596-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2596

Manuscript 1 (family 065, 2026-10-05): 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 propagation through projective bundles

OpenAI's claim

Full descendant Virasoro constraints are claimed to pass from any smooth projective base to projectivizations of arbitrary nonsplit algebraic bundles, including all genera, integral curve classes, and odd insertions. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For a smooth connected projective complex variety $B$, an algebraic vector bundle $E$ of rank $r\ge2$, and $X=\mathbb P_B(E)$ parametrizing lines in the fibers, OpenAI claims \[ \left[L_k^B Z_B=0\text{ for all }k\ge-1\right]\ \Longrightarrow\ \left[L_k^X Z_X=0\text{ for all }k\ge-1\right]. \] For each target $Y$, $Z_Y$ is its ordinary descendant partition function $\exp(\sum_g\hbar^{g-1}F_g^Y)$ with all cohomology classes, stable-map cotangent classes and actual effective integral curve labels. The normally ordered $L_k^Y$ use $\mu_Y|_{H^{p,q}}=p-\dim(Y)/2$, $R_Y=c_1(TY)\cup$, and zero-mode scalar $\chi(Y)/16-\operatorname{str}(\mu_Y^2)/4$; other modes have no scalar correction. The conclusion is coefficientwise in every genus, curve class and descendant list.

Outline of the argument

Manuscript 1 (family 065, 2026-10-05): Localization on the completion by a trivial summand separates the base and projective-bundle blocks. The logarithm of an equivariant shift cancels coefficient differentiation in the grading, giving quantizable modes; ancestor bounds continue their identities coefficientwise, and quantum Riemann--Roch identifies the ordinary constraints on both components.

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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