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arxiv: 2602.11002 · v2 · submitted 2026-02-11 · 🧮 math.DG · math.CV

Simply connectedness of K\"ahler and Riemannian manifolds via spectral estimates (with an appendix by Shiyu Zhang)

Francesco Bei This is my paper

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classification 🧮 math.DG math.CV
keywords Kähler manifoldssimply connectedrationally connectedspectral positivityRiemannian manifoldshomology spheresprojective manifoldstangent bundle slope
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The pith

Weak spectral positivity implies that compact Kähler manifolds are rationally connected and simply connected

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

The paper shows that a compact Kähler manifold satisfying a mild spectral positivity condition must be rationally connected. This forces the manifold to be simply connected and projective algebraic, with all holomorphic p-forms vanishing for p greater than zero. A parallel argument applies to compact oriented even-dimensional Riemannian manifolds, where an analogous spectral condition ensures they are simply connected real homology spheres. The results rest on deriving topological conclusions from lower bounds on the spectrum of appropriate operators. An appendix further characterizes the rational dimension of such Kähler manifolds by the positivity of the minimal slope of the tangent bundle.

Core claim

Under a rather weak spectral positivity assumption on a compact Kähler manifold (M,h), M is rationally connected and thus simply connected, projective with h^{p,0}(M)={0} for each p>0. For Riemannian manifolds, an appropriate spectral positivity assumption guarantees that a compact and oriented even-dimensional Riemannian manifold (M,g) is a simply connected real homology sphere. The appendix gives a characterization of the rational dimension of compact Kähler manifolds in terms of the positivity of the minimal slope of the tangent bundle.

What carries the argument

The rather weak spectral positivity assumption, which supplies lower bounds on eigenvalues or related spectral quantities that control connectivity and vanishing of cohomology

Load-bearing premise

The rather weak spectral positivity assumption holds for the compact Kähler or Riemannian manifold under consideration.

What would settle it

A compact Kähler manifold satisfying the spectral positivity assumption but having nonzero h^{p,0} for some p>0 or failing to be simply connected would disprove the claim.

read the original abstract

Let $(M,h)$ be a compact K\"ahler manifold. Under a rather weak spectral positivity assumption we prove that $M$ is rationally connected and thus simply connected, projective with $h^{p,0}(M)=\{0\}$ for each $p>0$. Then, in the second part of this paper, we focus on Riemannian manifolds and we provide an appropriate spectral positivity assumption which guarantees that a compact and oriented even dimensional Riemannian manifold $(M,g)$ is a simply connected real homology sphere. Finally, in the appendix, a characterization of the rational dimension of compact K\"ahler manifolds in terms of the positivity of the minimal slope of the tangent bundle is given.

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 / 2 minor

Summary. The manuscript proves that under a rather weak spectral positivity assumption, a compact Kähler manifold (M,h) is rationally connected (hence simply connected and projective) with h^{p,0}(M)=0 for all p>0. It extends the method to Riemannian manifolds, showing that a corresponding spectral positivity assumption on a compact oriented even-dimensional (M,g) implies it is a simply connected real homology sphere. The appendix characterizes the rational dimension of compact Kähler manifolds in terms of positivity of the minimal slope of the tangent bundle.

Significance. If the derivations hold, the results establish a direct analytic route from eigenvalue lower bounds to vanishing of holomorphic forms and rational connectedness via integrated Bochner formulas, providing new criteria for simple connectedness and projectivity. The appendix's slope-based characterization adds a bridge to stability notions in algebraic geometry. These implications are potentially impactful for classification problems in Kähler and Riemannian geometry.

minor comments (2)
  1. [§2] §2, Definition 2.1: Explicitly compare the spectral positivity lower bound to standard conditions (e.g., positive Ricci curvature or Bochner positivity) to substantiate the 'rather weak' descriptor.
  2. [Appendix] Appendix: Include a short remark relating the minimal-slope characterization to prior results on rational connectedness (e.g., those of Campana or Kollár) for context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. The summary accurately reflects the main theorems: weak spectral positivity implies rational connectedness (hence simple connectedness and projectivity) for compact Kähler manifolds, with vanishing of holomorphic forms, and simple connectedness as real homology spheres for even-dimensional Riemannian manifolds. The appendix's characterization via minimal slope is also correctly noted. We appreciate the recommendation for minor revision and will incorporate any editorial or minor clarifications in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds directly from the spectral positivity assumption (Definition 2.1) via integrated Bochner-Weitzenbock identities to vanishing of holomorphic forms, Hodge numbers, and rational connectedness. These steps rely on standard analytic estimates and do not reduce to parameter fitting, self-definitional loops, or load-bearing self-citations. The appendix characterization of rational dimension via minimal slope positivity is likewise derived from the same estimates without circular reduction to the main theorem's inputs. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background from Kähler geometry, Riemannian geometry, and spectral theory without introducing new free parameters or invented entities.

axioms (2)
  • standard math Standard definitions and properties of compact Kähler manifolds and their Hodge numbers
    Invoked throughout the first part to relate spectral positivity to rational connectedness and projectivity.
  • standard math Standard definitions and properties of compact oriented Riemannian manifolds and their homology
    Invoked in the second part to conclude simple connectedness and homology-sphere structure from spectral positivity.

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Reference graph

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