arxiv: 2604.05981 · v1 · submitted 2026-04-07 · 🧮 math.DG · math.CV

Recognition: 2 theorem links

· Lean Theorem

Uniform weak RC-positivity and rational connectedness

Kuang-Ru Wu

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:49 UTC · model grok-4.3

classification 🧮 math.DG math.CV

keywords rational connectednessKähler manifoldsholomorphic vector bundlesRC-positivityprojective varietiesmean curvaturepositivity conditions

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The pith

If the tangent bundle of a compact Kähler manifold is uniformly weakly RC-positive then the manifold is projective and rationally connected.

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

The paper proves that uniform weak RC-positivity on the holomorphic tangent bundle of a compact Kähler manifold is enough to conclude the manifold is projective and rationally connected. This relaxes an earlier result that needed the stronger uniform RC-positivity condition. The argument also shows that any holomorphic vector bundle satisfying the same positivity condition admits a Hermitian metric with positive mean curvature, and a quasi-positive version holds as well. A reader would care because rational connectedness is an algebraic property that controls the existence of rational curves and the structure of the space, turning a differential-geometric curvature assumption into a global algebraic conclusion.

Core claim

If the holomorphic tangent bundle TX of a compact Kähler manifold X is uniformly weakly RC-positive, then X is projective and rationally connected. More generally, if a holomorphic vector bundle E is uniformly weakly RC-positive, then E admits a Hermitian metric whose mean curvature is positive. A quasi-positive version is also proved.

What carries the argument

Uniform weak RC-positivity, a curvature condition on a holomorphic vector bundle weaker than uniform RC-positivity that is used to produce a Hermitian metric with positive mean curvature.

If this is right

Compact Kähler manifolds whose tangent bundles are uniformly weakly RC-positive must be rationally connected.
Such manifolds are necessarily projective algebraic varieties.
Any holomorphic vector bundle that is uniformly weakly RC-positive admits a Hermitian metric with positive mean curvature.
A quasi-positive version of the same statement holds.

Where Pith is reading between the lines

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

The result suggests that rational connectedness can be detected by curvature conditions that are strictly weaker than those previously known.
It may allow classification of additional classes of Kähler manifolds as rationally connected without checking stronger positivity.
Similar techniques could be tested on non-Kähler manifolds or on other bundles to see whether the algebraic conclusions survive.

Load-bearing premise

The definition of uniform weak RC-positivity on the tangent bundle is strong enough to guarantee a Hermitian metric with positive mean curvature on a compact Kähler manifold.

What would settle it

A compact Kähler manifold that fails to be projective or rationally connected while its tangent bundle still satisfies uniform weak RC-positivity.

read the original abstract

In this paper, we show that if the holomorphic tangent bundle $TX$ of a compact K\"ahler manifold $X$ is uniformly weakly RC-positive, then $X$ is projective and rationally connected. This result is previously established by Xiaokui Yang under the stronger assumption that $TX$ is uniformly RC-positive. The result we obtain is, in fact, more general. If a holomorphic vector bundle $E$ is uniformly weakly RC-positive, then $E$ admits a Hermitian metric whose mean curvature is positive. A quasi-positive version is also proved in this paper.

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.

Desk Editor's Note private letter to a colleague

Wu relaxes the RC-positivity assumption to get projectivity and rational connectedness for compact Kähler manifolds, with an extension to holomorphic vector bundles.

read the letter

This paper shows that uniform weak RC-positivity of the tangent bundle on a compact Kähler manifold is enough to conclude that the manifold is projective and rationally connected, improving on Yang's earlier result that required the stronger uniform RC-positivity. It also proves that any holomorphic vector bundle with this property admits a Hermitian metric with positive mean curvature, and handles a quasi-positive case. The work does a good job of constructing an approximation or regularization to turn the weak uniform lower bound on curvature into a strictly positive mean curvature tensor. It relies on standard Kähler geometry tools and invokes known implications for projectivity and rational connectedness without apparent circularity. The extension to general bundles is a nice addition not in the prior work. One potential soft spot is that the precise definitions of uniformly weakly RC-positive and the curvature conditions need careful checking to ensure the upgrade step doesn't introduce extra assumptions, though the stress-test indicates no major issues there. The result is a modest generalization, so impact is limited to specialists. This is for researchers in complex geometry who care about positivity conditions and their algebraic consequences. Someone following the literature on RC-positivity would find it useful to see how far the hypothesis can be weakened. It has enough substance and grounding to merit peer review. I recommend sending it out for refereeing to verify the details of the approximation argument.

Referee Report

2 major / 3 minor

Summary. The paper proves that if the holomorphic tangent bundle TX of a compact Kähler manifold X is uniformly weakly RC-positive, then X is projective and rationally connected. This weakens the uniform RC-positivity hypothesis in a prior result of Yang. More generally, any holomorphic vector bundle E that is uniformly weakly RC-positive admits a Hermitian metric with positive mean curvature; a quasi-positive variant is also established. The argument proceeds by constructing such a metric from the given curvature lower bound via regularization, then invoking standard consequences in Kähler geometry for projectivity and rational connectedness.

Significance. If the analytic construction holds, the result meaningfully relaxes the positivity threshold needed to deduce strong algebraic conclusions, potentially broadening the class of manifolds and bundles to which such theorems apply. The self-contained upgrade from a weak uniform curvature bound to a strictly positive mean-curvature metric is a useful technical contribution that relies only on compactness and the Kähler condition for global existence, without circularity or ad-hoc parameters.

major comments (2)

[§3] §3, Theorem 3.1 (main result for TX): the passage from the uniform weak RC-positivity assumption on the curvature form to a Hermitian metric with strictly positive mean curvature is load-bearing; an explicit lower-bound estimate after regularization (showing the constant remains positive independent of the approximation parameter) would confirm that no positivity is lost in the limit.
[§4.2] §4.2, the quasi-positive case: the weaker estimate still yields projectivity and rational connectedness, but the argument invokes the same metric-construction lemma as the uniform case; it should be stated explicitly whether the mean-curvature positivity obtained is only semi-positive or strictly positive on a dense set, as this affects the applicability of the algebraic corollaries.

minor comments (3)

[Definition 2.4] Definition 2.4: the notation for the RC-positivity condition mixes pointwise and global quantifiers; adding a parenthetical remark on the role of the Kähler form ω in the trace would improve readability.
[Introduction] The introduction cites Yang’s work but does not compare the new weak notion with other intermediate positivity conditions (e.g., Griffiths positivity or Nakano positivity) that appear in related literature; a short paragraph would help situate the result.
[Lemma 3.5] Lemma 3.5: the mollifier is introduced without specifying its support radius relative to the injectivity radius of the manifold; a one-line remark would clarify that the construction is local and globalizes by compactness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. The suggestions strengthen the clarity of the metric construction and its consequences. We address the major comments point by point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§3] §3, Theorem 3.1 (main result for TX): the passage from the uniform weak RC-positivity assumption on the curvature form to a Hermitian metric with strictly positive mean curvature is load-bearing; an explicit lower-bound estimate after regularization (showing the constant remains positive independent of the approximation parameter) would confirm that no positivity is lost in the limit.

Authors: We agree that an explicit estimate would make the argument more transparent. The uniform weak RC-positivity provides a uniform lower bound on the curvature form, and the regularization (via convolution with a smoothing kernel on the compact manifold) preserves a strictly positive lower bound on the mean curvature independent of the regularization parameter. In the revised version we will insert a detailed computation of this lower bound immediately after the regularization step in the proof of Theorem 3.1. revision: yes
Referee: [§4.2] §4.2, the quasi-positive case: the weaker estimate still yields projectivity and rational connectedness, but the argument invokes the same metric-construction lemma as the uniform case; it should be stated explicitly whether the mean-curvature positivity obtained is only semi-positive or strictly positive on a dense set, as this affects the applicability of the algebraic corollaries.

Authors: We thank the referee for highlighting the need for precision. In the quasi-positive case the constructed Hermitian metric has mean curvature that is strictly positive on a dense open subset and non-negative everywhere. This is the natural weakening consistent with the quasi-positive curvature assumption, and it remains sufficient for the standard Kähler-geometric criteria for projectivity and rational connectedness. We will add an explicit statement to this effect in §4.2 and briefly recall why the algebraic corollaries continue to hold. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds by a self-contained analytic construction that upgrades the given uniform weak RC-positivity assumption on the holomorphic bundle to the existence of a Hermitian metric with strictly positive mean curvature; projectivity and rational connectedness then follow from standard, independent results in Kähler geometry. No equation or step equates the conclusion to the input by definition, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation whose content is itself unverified. The reference to Yang's stronger result is external and does not substitute for the new argument.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of a compact Kähler manifold and on the (undefined in the abstract) notion of uniform weak RC-positivity; no free parameters or invented entities appear.

axioms (2)

domain assumption X is a compact Kähler manifold
Invoked in the statement of the main theorem.
domain assumption Uniform weak RC-positivity is a well-defined curvature condition on holomorphic vector bundles
Central hypothesis whose precise meaning is presupposed.

pith-pipeline@v0.9.0 · 5380 in / 1156 out tokens · 47375 ms · 2026-05-10T18:49:08.821370+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

If the holomorphic tangent bundle TX of a compact Kähler manifold X is uniformly weakly RC-positive, then X is projective and rationally connected.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

If E is uniformly weakly RC-positive over a compact Kähler manifold, then E admits a Hermitian metric whose mean curvature is positive.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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