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On the local topology of non-collapsed Ricci bounded limit spaces

Song Sun, Jikang Wang, Junsheng Zhang

TL;DR

This work proves that for pointed Gromov–Hausdorff limits of non-collapsed manifolds with bounded Ricci curvature, the local $b_1$ of the regular locus vanishes by establishing a uniform exponent bound on the image of $H_1$ between small and unit balls. The authors develop an effective regular set framework, prove a weak main theorem via loop decomposition and an Anderson-type regularity argument, and then upgrade to a full exponent bound through a rescaling inductive scheme on the volume ratio $v$. They derive significant applications to Kähler geometry, showing log terminal singularities for cones and polarized limits, and obtain global exponent bounds for $H_1$ on regular parts of Ricci-limit spaces, with further discussion of open problems and potential extensions to complex-analytic and analytic-regular settings. The results provide a robust topological control on the regular loci of non-collapsed Ricci limit spaces and pave the way for deeper connections between metric geometry and complex-analytic singularity theory.

Abstract

We show that for a pointed Gromov-Hausdorff limit of non-collapsed Riemannian manifolds with bounded Ricci curvature, the local $b_1$ of the regular loci vanishes. We also discuss applications and some open questions.

On the local topology of non-collapsed Ricci bounded limit spaces

TL;DR

This work proves that for pointed Gromov–Hausdorff limits of non-collapsed manifolds with bounded Ricci curvature, the local of the regular locus vanishes by establishing a uniform exponent bound on the image of between small and unit balls. The authors develop an effective regular set framework, prove a weak main theorem via loop decomposition and an Anderson-type regularity argument, and then upgrade to a full exponent bound through a rescaling inductive scheme on the volume ratio . They derive significant applications to Kähler geometry, showing log terminal singularities for cones and polarized limits, and obtain global exponent bounds for on regular parts of Ricci-limit spaces, with further discussion of open problems and potential extensions to complex-analytic and analytic-regular settings. The results provide a robust topological control on the regular loci of non-collapsed Ricci limit spaces and pave the way for deeper connections between metric geometry and complex-analytic singularity theory.

Abstract

We show that for a pointed Gromov-Hausdorff limit of non-collapsed Riemannian manifolds with bounded Ricci curvature, the local of the regular loci vanishes. We also discuss applications and some open questions.
Paper Structure (5 sections, 16 theorems, 89 equations)

This paper contains 5 sections, 16 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the main theorem
  4. Applications
  5. Discussion

Key Result

Theorem 1.1

Given $v \in (0,1]$ and $n \in \mathbb{Z}_+$, there exist constants $t= t(n, v) > 0$ and $C = C(n, v) > 0$ such that for any $(X, d, p) \in \mathcal{M}(n, v)$, the local homology group has exponent bounded by $C$.

Theorems & Definitions (36)

  • Theorem 1.1: Exponent bound of local homology group
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 2.1: A90CC97
  • Theorem 2.2: CN15
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 26 more