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Math-2341 · mathematics · Algebraic geometry

Algebraic Dimension and Reduction of Hyperkähler 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.
Let $X$ be a hyperkähler manifold of complex dimension $2n$. Is its algebraic dimension necessarily in $\{0,n,2n\}$? If $a(X)=n$, does the algebraic reduction admit a holomorphic model $f:X\to B$, where $B$ is a normal projective variety of dimension $n$, and hence define a Lagrangian fibration? Does parabolic $\operatorname{NS}(X)$ imply $a(X)=n$?

Formulation references: [S1], [S2].

Short English statement

Let $X$ be a hyperkähler manifold of complex dimension $2n$. Is its algebraic dimension necessarily in $\{0,n,2n\}$? If $a(X)=n$, does the algebraic reduction admit a holomorphic model $f:X\to B$, where $B$ is a normal projective variety of dimension $n$, and hence define a Lagrangian fibration? Does parabolic $\operatorname{NS}(X)$ imply $a(X)=n$?

Sources

Catalogue entry

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

Contributions 1

Math-2341-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2341

Manuscript 1 (family 041, 2026-09-23): A nonzero nef isotropic line produces a Lagrangian fibration after semiampleness, providing the expected intermediate algebraic reduction in that additional-line class. The record asks the algebraic-dimension classification for arbitrary hyperkahler manifolds without this nef-line hypothesis.

Algebraic Dimension and Reduction of Hyperkähler Manifolds

OpenAI's claim

On a compact Kähler IHS manifold, a holomorphic line is claimed to be semiample whenever its Chern class is nonzero, nef, and isotropic, regardless of deformation type. 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 $X$ be an irreducible holomorphic symplectic manifold, required to be compact and Kähler, of complex dimension $2n$, $n\ge1$. Write $q_X$ for its Beauville--Bogomolov form and $\overline{\mathcal K_X}$ for the closed Kähler cone. For a holomorphic line bundle $L$ with nonzero $c_1(L)\in\overline{\mathcal K_X}$ and $q_X(c_1(L))=0$, OpenAI claims semiampleness and, more precisely, \[ L^m\simeq f^*A,\qquad f:X\twoheadrightarrow B,\qquad\dim_{\mathbb C}B=n, \] with $m>0$, connected fibers, normal projective $B$ and ample $A$. Every fiber component has dimension $n$ and the holomorphic symplectic form vanishes on its smooth locus. A separate corollary claims finitely many smooth deformation types in each dimension $2n$ among these manifolds with $b_2\ge5$.

Outline of the argument

Manuscript 1 (family 041, 2026-09-23): The prescribed lattice vector is retained through a degeneration that produces a special Lagrangian torus. Rotation makes that torus holomorphic on a nonprojective deformation, where the torus-fibration theorem supplies semiampleness; the fixed-class deformation result brings the conclusion back to the original manifold.

Formal verification

Family 041 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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