Bases of projective Lagrangian fibrations
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
For $n\geq 1$, let X be a compact irreducible holomorphic symplectic Kähler manifold of dimension 2n and $f:X\to B$ a projective surjective holomorphic map with connected fibers to a normal projective n-fold. If every irreducible component of every fiber has dimension n and its smooth locus is isotropic for the symplectic form, must B be isomorphic to $P^n$?Short English statement
For $n\geq 1$, let X be a compact irreducible holomorphic symplectic Kähler manifold of dimension 2n and $f:X\to B$ a projective surjective holomorphic map with connected fibers to a normal projective n-fold. If every irreducible component of every fiber has dimension n and its smooth locus is isotropic for the symplectic form, must B be isomorphic to $P^n$?Sources
- S1 Original question source or mathematical context. https://arxiv.org/abs/0804.1682.
- S2 Original question source or mathematical context. https://doi.org/10.4171/owr/2008/38.
- S3 Original question source or mathematical context. https://epiga.episciences.org/17061/pdf.
- S4 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2575
Manuscript 1 (family 041, 2026-09-23): 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.
Bases of projective Lagrangian fibrations
OpenAI's claim
Every projective Lagrangian fibration of a compact Kähler IHS manifold is claimed to have projective space as its base. 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 $(X,\sigma)$ be a compact Kähler irreducible holomorphic symplectic manifold of dimension $2n$, with $n\ge1$. Assume a projective connected-fiber surjective holomorphic morphism $f:X\to B$ to normal projective $B$ of dimension $n$, with every fiber component of dimension $n$ and $\sigma$ zero on each component's smooth locus. OpenAI claims \[ B\simeq\mathbb P^n_{\mathbb C}. \] The base need not initially be smooth; the hypotheses include projectivity of the morphism and the componentwise Lagrangian condition on every fiber.Outline of the argument
Manuscript 1 (family 041, 2026-09-23): The first stage excludes nonquotient singularities using fixed-base deformations, Euler obstructions, and Hodge variation. A finite abelian cover and a second varying fibration eliminate the residual quotient stabilizers, establishing smoothness before the known smooth-base theorem is applied.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
- OpenAI, Projective-space bases of Lagrangian fibrations, 2026-09-23; citation key
OAI:Projective-Space-Bases-of-Lagrangian-Fibrations-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.