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Orbifold modifications of complex analytic spaces

János Kollár, Wenhao Ou

TL;DR

The paper proves that a compact complex analytic space with a locally trivial orbifold structure along a dense singular subset admits a projective, bimeromorphic orbifold modification Y→X that is an isomorphism over the locally trivial locus and the smooth locus. It advances Xu's projective result to the analytic setting under local-triviality assumptions by a four-step construction: (1) well-prepared neighborhoods to control equisingularity, (2) constrained resolutions isolating the problematic locus, (3) local chart resolutions yielding quotient-singularity models, and (4) patching these local models into a global modification. The approach relies on carefully chosen finite covers and $H$-equivariant resolutions to produce snc configurations and to glue local data compatibly. The authors also discuss the open analytic case without local triviality and provide illustrative examples highlighting the method's nuances and limitations.

Abstract

We show that a compact, complex analytic space $X$ has a bimeromorphic orbifold modification that is an isomorphism over the locally trivial orbifold locus of $X$.

Orbifold modifications of complex analytic spaces

TL;DR

The paper proves that a compact complex analytic space with a locally trivial orbifold structure along a dense singular subset admits a projective, bimeromorphic orbifold modification Y→X that is an isomorphism over the locally trivial locus and the smooth locus. It advances Xu's projective result to the analytic setting under local-triviality assumptions by a four-step construction: (1) well-prepared neighborhoods to control equisingularity, (2) constrained resolutions isolating the problematic locus, (3) local chart resolutions yielding quotient-singularity models, and (4) patching these local models into a global modification. The approach relies on carefully chosen finite covers and -equivariant resolutions to produce snc configurations and to glue local data compatibly. The authors also discuss the open analytic case without local triviality and provide illustrative examples highlighting the method's nuances and limitations.

Abstract

We show that a compact, complex analytic space has a bimeromorphic orbifold modification that is an isomorphism over the locally trivial orbifold locus of .
Paper Structure (4 sections, 6 theorems, 22 equations)

This paper contains 4 sections, 6 theorems, 22 equations.

Table of Contents

  1. Well prepared neighborhoods
  2. Constrained resolutions
  3. Resolution of local charts
  4. Patching the local charts

Key Result

Theorem 1

Let $X$ be a compact, complex space and $S^\circ\subset \operatorname{Sing}(X)$ a Zariski open subset such that $X$ is a locally trivial orbifold along $S^\circ$. Then there is a projective, bimeromorphic modification $\pi_Y:Y\to X$, such that $Y$ is an orbifold, and $\pi_Y$ is an isomorphism over b

Theorems & Definitions (10)

  • Theorem 1
  • Example 3
  • Example 7
  • Proposition 9
  • Lemma 10
  • Definition 11
  • Lemma 12
  • Proposition 13
  • Remark 14
  • Lemma 16