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arxiv: 2205.07958 · v4 · pith:POZDENEGnew · submitted 2022-05-16 · 🧮 math.AG · math.CV

On algebraically coisotropic submanifolds of holomorphic symplectic manifolds

Ekaterina Amerik , Fr\'ed\'eric Campana This is my paper

Pith reviewed 2026-05-24 12:07 UTC · model grok-4.3

classification 🧮 math.AG math.CV
keywords algebraically coisotropicholomorphic symplecticLagrangian submanifoldabelian varietynef canonical bundleuniruledétale coverproduct decomposition
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The pith

Non-uniruled algebraically coisotropic submanifolds decompose into Lagrangian products after finite cover.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines algebraically coisotropic submanifolds X inside holomorphic symplectic projective manifolds M. It asks if a non-uniruled X makes the pair (X,M) a product (Z×Y, N×Y) up to finite étale cover, with Z Lagrangian in the holomorphic symplectic N. This holds when M is an abelian variety. When the canonical bundle K_X is nef and big, X is Lagrangian in M, and the same is true without the nefness condition.

Core claim

When X is a non-uniruled algebraically coisotropic submanifold of a holomorphic symplectic projective manifold M and M is an abelian variety, then up to a finite étale cover the pair (X,M) is a product (Z×Y, N×Y) where N and Y are holomorphic symplectic and Z is Lagrangian in N. In particular, if K_X is nef and big then X is Lagrangian in M, and this conclusion is valid even without assuming nefness. Lagrangian submanifolds do not exist on a sufficiently general abelian variety.

What carries the argument

The product decomposition of the pair (X,M) into a Lagrangian factor Z in N and a holomorphic symplectic factor Y, triggered by the non-uniruled assumption on X.

If this is right

  • The decomposition holds for all non-uniruled algebraically coisotropic submanifolds when the ambient manifold is abelian.
  • X must be Lagrangian in M whenever K_X is nef and big.
  • The Lagrangian conclusion does not require the nefness of K_X.
  • Lagrangian submanifolds cannot exist inside sufficiently general abelian varieties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar product structures could be sought in irreducible hyperkähler manifolds.
  • The non-existence result differentiates abelian varieties from hyperkähler manifolds regarding possible submanifolds.
  • The semi-ampleness case might be strengthened to the full decomposition result with further work.

Load-bearing premise

X is assumed to be algebraically coisotropic in the holomorphic symplectic projective manifold M.

What would settle it

An example of a non-uniruled algebraically coisotropic submanifold X in an abelian variety M that does not admit the product decomposition (Z×Y, N×Y) after any finite étale cover.

read the original abstract

We investigate algebraically coisotropic submanifolds $X$ in a holomorphic symplectic projective manifold $M$. Motivated by our results in the hypersurface case, we raise the following question: when $X$ is not uniruled, is it true that up to a finite \'etale cover, the pair $(X,M)$ is a product $(Z\times Y, N\times Y)$ where $N, Y$ are holomorphic symplectic and $Z\subset N$ is Lagrangian? We prove that this is indeed the case when $M$ is an abelian variety, and give some partial answer when the canonical bundle $K_X$ is semi-ample. In particular, when $K_X$ is nef and big, $X$ is Lagrangian in $M$ (in fact this also holds without nefness assumption). We also remark that Lagrangian submanifolds do not exist on a sufficiently general Abelian variety, in contrast to the case when $M$ is irreducible hyperk\"ahler.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies algebraically coisotropic submanifolds X inside holomorphic symplectic projective manifolds M. Motivated by the hypersurface case, it asks whether a non-uniruled X yields, up to finite étale cover, a product decomposition (X,M) ≅ (Z×Y, N×Y) with Z Lagrangian in the holomorphic symplectic manifold N. The authors prove the decomposition when M is an abelian variety, obtain partial results when K_X is semi-ample, and show in particular that K_X nef and big forces X to be Lagrangian in M (the nef hypothesis is unnecessary). They also remark that sufficiently general abelian varieties carry no Lagrangian submanifolds, in contrast to the irreducible hyperkähler case.

Significance. If the stated theorems hold, the work supplies a structural decomposition theorem for non-uniruled algebraically coisotropic submanifolds in the abelian case and a clean positivity criterion forcing the Lagrangian property. The contrast between abelian and irreducible hyperkähler ambient spaces is a useful clarification. The results rest on standard tools of holomorphic symplectic geometry and algebraic geometry rather than ad-hoc constructions.

minor comments (3)
  1. The abstract states that the nef hypothesis is unnecessary for the Lagrangian conclusion when K_X is big; the corresponding theorem statement in the body should make the precise minimal hypothesis explicit (e.g., “K_X big” alone) and indicate where nefness was used in an intermediate step that can be removed.
  2. Notation for the finite étale cover and the factors N, Y, Z should be introduced once in a preliminary section and used consistently; currently the product decomposition is described in slightly varying language between the question and the theorem statements.
  3. The remark on the non-existence of Lagrangian submanifolds on a general abelian variety would benefit from a brief reference or short argument sketch, even if it is not the main focus.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation of minor revision. No major comments were raised in the report, so we have no specific points to address point-by-point. We will make any minor editorial changes suggested by the editor or referee in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central claims (product decomposition for abelian M when X is non-uniruled, and X Lagrangian when K_X is nef and big or semi-ample) are established via direct case analysis drawing on standard facts about holomorphic symplectic manifolds, canonical bundles, and algebraic geometry. The non-uniruled hypothesis serves only to trigger the decomposition result the authors prove in the abelian case; the hypersurface motivation is cited only for context and does not enter the derivations. No step reduces by definition, by fitted-parameter renaming, or by a self-citation chain that replaces an independent argument. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard background theorems in complex algebraic geometry; no free parameters, ad-hoc axioms, or new entities are introduced.

axioms (1)
  • standard math Standard properties of holomorphic symplectic manifolds, projective varieties, and the canonical bundle in algebraic geometry
    Invoked throughout to define coisotropic, Lagrangian, and nef/big conditions.

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Reference graph

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