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arxiv: 2001.10654 · v4 · pith:GDD4YATGnew · submitted 2020-01-29 · 🧮 math.AG · math.CV

On the dual positive cones and the algebraicity of a compact K\"ahler manifold

Hsueh-Yung Lin This is my paper

Pith reviewed 2026-05-24 15:07 UTC · model grok-4.3

classification 🧮 math.AG math.CV
keywords compact Kähler manifolddual Kähler conerational classAlbanese varietyprojectivityalgebraicitythreefoldspositive cone
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The pith

If the dual Kähler cone of a compact Kähler manifold contains a rational class as an interior point, then its Albanese variety is projective.

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

The paper shows that when a rational class sits in the interior of the dual Kähler cone of a compact Kähler manifold, the Albanese variety must be projective. This conclusion is reached for manifolds that admit a positive rational Hodge class of bidimension (1,1). If the statement holds, the location of rational classes inside the dual cone directly determines an algebraic feature of the manifold. The argument also yields algebraicity statements for Ricci-flat cases and examines the situation in dimension three.

Core claim

If the dual Kähler cone of a compact Kähler manifold X contains a rational class as an interior point, then its Albanese variety is projective. This supplies a direct link between the rational points of the dual cone and the projectivity of the Albanese variety.

What carries the argument

The dual Kähler cone together with the condition that a rational class lies in its interior, which forces the Albanese variety to be projective.

If this is right

  • The Albanese variety of any such manifold is projective.
  • Ricci-flat compact Kähler manifolds that satisfy the cone condition have projective Albanese varieties.
  • Analogous algebraicity conclusions hold when the manifold is a threefold.

Where Pith is reading between the lines

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

  • The cone condition supplies a practical test for projectivity of the Albanese variety on explicit Kähler examples.
  • The same criterion may relate to broader questions about which Kähler manifolds admit algebraic structures.
  • One could check whether the converse implication holds under mild extra hypotheses on the manifold.

Load-bearing premise

The compact Kähler manifold admits a positive rational Hodge class of bidimension (1,1).

What would settle it

A compact Kähler manifold in which a rational class lies in the interior of the dual Kähler cone yet whose Albanese variety fails to be projective would disprove the claim.

read the original abstract

We investigate the algebraicity of compact K\"ahler manifolds admitting a positive rational Hodge class of bidimension $(1,1)$. We prove that if the dual K\"ahler cone of a compact K\"ahler manifold $X$ contains a rational class as an interior point, then its Albanese variety is projective. As a consequence, we answer the Oguiso--Peternell problem for Ricci-flat compact K\"ahler manifolds. We also study related algebraicity problems for threefolds.

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

2 major / 2 minor

Summary. The paper studies algebraicity questions for compact Kähler manifolds that admit a positive rational Hodge class of bidimension (1,1). Its central result states that if the dual Kähler cone of such an X contains a rational class in its interior, then the Albanese variety of X is projective. As a corollary the Oguiso–Peternell question is settled for Ricci-flat compact Kähler manifolds; the paper also treats related algebraicity problems in dimension three.

Significance. If the main implication holds, the work supplies a concrete numerical criterion (rational interior point in the dual Kähler cone) that forces projectivity of the Albanese variety, thereby giving a new handle on the algebraicity problem for Kähler manifolds. The resolution of the Oguiso–Peternell conjecture in the Ricci-flat case is a concrete advance in the field.

major comments (2)
  1. [Theorem 1.1 / §2] Theorem 1.1 (or the statement labeled as the main result in §2): the argument that a rational interior point forces the Albanese map to be algebraic appears to rest on the existence of a positive (1,1)-class that is rational; it is not immediately clear from the sketch whether this class is used only to guarantee the interior point or whether an additional positivity assumption is needed to close the argument. A precise reference to the lemma that converts the interior-point hypothesis into an algebraic cycle would clarify the logical dependence.
  2. [§4] §4 (the three-dimensional case): the reduction to the case where the dual cone contains a rational interior point is stated to follow from the main theorem, yet the verification that the Hodge class remains positive after the reduction step is only indicated and not written out. If this step uses a deformation argument, the deformation space and the openness of the interior-point condition should be made explicit.
minor comments (2)
  1. [Introduction] The notation for the dual Kähler cone (often denoted K^∨ or similar) is introduced without a displayed definition in the introduction; adding an explicit sentence or equation would help readers who are not already familiar with the dual-cone literature.
  2. [Introduction] Several citations to works on the Oguiso–Peternell problem appear only in the bibliography; a short sentence in the introduction recalling the precise statement of the problem would make the motivation self-contained.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise comments, which help clarify the logical structure of the arguments. We address each point below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Theorem 1.1 / §2] Theorem 1.1 (or the statement labeled as the main result in §2): the argument that a rational interior point forces the Albanese map to be algebraic appears to rest on the existence of a positive (1,1)-class that is rational; it is not immediately clear from the sketch whether this class is used only to guarantee the interior point or whether an additional positivity assumption is needed to close the argument. A precise reference to the lemma that converts the interior-point hypothesis into an algebraic cycle would clarify the logical dependence.

    Authors: The rational class in the interior of the dual Kähler cone is the positive rational Hodge class of bidimension (1,1) from the setup of the paper. This single class is used both to satisfy the interior-point hypothesis and to apply the algebraic-cycle correspondence. We have inserted an explicit cross-reference to Lemma 2.4 (which converts an interior rational point into an algebraic cycle via the Hodge conjecture in this setting) at the relevant step of the proof of Theorem 1.1. No further positivity assumption is imposed beyond the interior-point condition itself. revision: yes

  2. Referee: [§4] §4 (the three-dimensional case): the reduction to the case where the dual cone contains a rational interior point is stated to follow from the main theorem, yet the verification that the Hodge class remains positive after the reduction step is only indicated and not written out. If this step uses a deformation argument, the deformation space and the openness of the interior-point condition should be made explicit.

    Authors: The reduction in §4 is indeed by deformation within the Kähler moduli space of the threefold. We have expanded the paragraph to state explicitly that the deformation space is the local moduli space of Kähler structures (smooth by the standard Kuranishi theory for Kähler manifolds) and that both the positivity of the Hodge class and the interior-point condition are open in the space of (1,1)-classes; hence they persist for a nearby deformation. The rewritten argument now contains the full verification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The abstract states a theorem relating the dual Kähler cone containing a rational interior point to projectivity of the Albanese variety, with no equations, fitted parameters, or self-citations visible in the supplied text. No load-bearing step reduces by construction to its own inputs, and the central implication is presented as a proved result rather than a renaming or self-referential definition. This matches the default expectation for papers without exhibited circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Paper relies on standard background from Hodge theory and Kähler geometry; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • standard math Standard properties of Hodge classes, Kähler cones, and Albanese varieties on compact Kähler manifolds
    Invoked implicitly as the setting for the algebraicity investigation.

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