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Math-2434 · mathematics · Algebraic geometry

Universal-Cover Decomposition for Nef Anticanonical Manifolds

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
If $X$ is a compact Kähler manifold with nef anticanonical class, does its universal cover decompose as

$$\widetilde X\simeq\mathbb C^q\times\prod_jY_j\times\prod_kS_k\times Z,$$

where each $Y_j$ is an irreducible Calabi–Yau manifold, each $S_k$ is an irreducible hyperkähler manifold, and $Z$ is rationally connected?

Formulation references: [S1], [S2].

Short English statement

If $X$ is a compact Kähler manifold with nef anticanonical class, does its universal cover decompose as

$$\widetilde X\simeq\mathbb C^q\times\prod_jY_j\times\prod_kS_k\times Z,$$

where each $Y_j$ is an irreducible Calabi–Yau manifold, each $S_k$ is an irreducible hyperkähler manifold, and $Z$ is rationally connected?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2434-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2434

Manuscript 1 (family 067, 2026-09-25): The nef-tangent uniformization corollary proves the claimed product form in its dimension-at-most-six or bounded Albanese-fiber class. The problem requires arbitrary compact Kahler manifolds with merely nef anticanonical line, not nef tangent and these dimension restrictions.

Universal-Cover Decomposition for Nef Anticanonical Manifolds

OpenAI's claim

Among smooth complex Fano varieties in dimension six, those with nef tangent are claimed to be rational homogeneous, with a universal-cover consequence when dimension minus maximal étale irregularity is at most six. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims rational homogeneity for each smooth connected complex projective Fano manifold of dimension six whose tangent bundle is nef (a quotient of a complex semisimple group by a parabolic subgroup). Its Kähler uniformization corollary considers a compact connected Kähler $X$ with analytically nef tangent bundle and \[ 0\le d=\dim X-\widetilde q(X)\le6,\qquad \widetilde q(X)=\max_{X'\to X\text{ finite étale}}h^1(X',\mathcal O_{X'}). \] It claims a connected finite étale cover whose Albanese morphism is a holomorphically locally trivial bundle with rational homogeneous fiber $F$ of dimension $d$, and \[ \widetilde X\simeq F\times\mathbb C^{\widetilde q(X)}. \] Thus the universal cover is quasi-projective and holomorphically convex, with Remmert reduction to the affine factor. In particular the corollary applies in complex dimension at most six.

Outline of the argument

Manuscript 1 (family 067, 2026-09-25): Known contractions reduce the new case to Picard number one and pseudoindices four or five. The universal minimal-curve family has two smooth fibrations whose projection identities give linear intersection constraints; characteristic-number and integrality comparisons rule out the remaining models and identify the homogeneous possibilities.

Formal verification

Family 067 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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