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Math-2567 · mathematics · Algebraic geometry

Polynomiality of the natural invariant anticanonical index

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let a compact real torus T act holomorphically on a compact complex manifold F and linearize $L=-K_F$ by the differential action. Put $I_T(m)=\Sigma _q(-1)^q$ dim $H^q(F,L^m)^T$. Must some positive a and rational polynomial P satisfy $I_T(m)=P(m)$ for every positive multiple of a, with $P(0)=I_T(0)$? No positivity or projectivity is assumed.

Short English statement

Let a compact real torus T act holomorphically on a compact complex manifold F and linearize $L=-K_F$ by the differential action. Put $I_T(m)=\Sigma _q(-1)^q$ dim $H^q(F,L^m)^T$. Must some positive a and rational polynomial P satisfy $I_T(m)=P(m)$ for every positive multiple of a, with $P(0)=I_T(0)$? No positivity or projectivity is assumed.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2567-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2567

Manuscript 1 (family 068, 2026-09-26): 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.

Polynomiality of the natural invariant anticanonical index

OpenAI's claim

The paper claims a polynomial invariant Euler index on divisible anticanonical exponents and a conversion that removes a fixed pseudo-effective error from unbounded effective anticanonical twists. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Consider a holomorphic action by a compact real torus $T$ on a compact complex manifold $F$, with natural differential linearization on $L_F=-K_F$. Define \[ I_T(m)=\sum_{q\ge0}(-1)^q\dim H^q(F,L_F^m)^T. \] OpenAI claims some integer $a>0$ and $P\in\mathbb Q[t]$ with $I_T(m)=P(m)$ for every positive multiple of $a$, and $P(0)=I_T(0)$. Its independent conversion theorem takes smooth connected projective complex $S$, smoothly semipositive $L=-K_S$ and pseudo-effective Cartier $D$; sections $H^0(S,m_jL-D)\ne0$ for a strictly increasing positive sequence $m_j$ are claimed to imply $H^0(S,kL)\ne0$ for some $k>0$. This conversion statement has no group-action hypothesis.

Outline of the argument

Manuscript 1 (family 068, 2026-09-26): Lefschetz denominators become weighted lattice counts with selected faces deleted, and a polyhedral retraction computes their value at exponent zero. For conversion, a maximal rational base makes section ratios descend to base functions and forces affine horizontal vanishing orders, permitting descent of forms before the fixed error is removed.

Formal verification

Family 068 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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