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On quasi-log structures for complex analytic spaces

Osamu Fujino

TL;DR

This work develops the analytic analogue of Ambro’s quasi-log framework by introducing quasi-log complex analytic spaces and proving foundational results such as adjunction and vanishing theorems in this setting. It then establishes a robust basepoint-free theory, Reid–Fukuda-type results, effective freeness, and cone/ contraction theorems for quasi-log spaces, enabling a minimal model-program–style toolkit for projective morphisms between complex analytic spaces. A key achievement is showing that analytic semi-log canonical pairs carry natural quasi-log structures, which furnishes vanishing, adjunction, and basepoint-freeness results for slc geometry in the analytic category. Collectively, these results extend the Mori program to highly singular complex analytic spaces and provide a unified framework to study slc and quasi-log phenomena analytically, with explicit bounds and constructive steps for generation and very ampleness.

Abstract

We introduce the notion of quasi-log complex analytic spaces and establish various fundamental properties. Moreover, we prove that a semi-log canonical pair naturally has a quasi-log complex analytic space structure. This paper is part of the author's project to establish a minimal model theory for projective morphisms between complex analytic spaces.

On quasi-log structures for complex analytic spaces

TL;DR

This work develops the analytic analogue of Ambro’s quasi-log framework by introducing quasi-log complex analytic spaces and proving foundational results such as adjunction and vanishing theorems in this setting. It then establishes a robust basepoint-free theory, Reid–Fukuda-type results, effective freeness, and cone/ contraction theorems for quasi-log spaces, enabling a minimal model-program–style toolkit for projective morphisms between complex analytic spaces. A key achievement is showing that analytic semi-log canonical pairs carry natural quasi-log structures, which furnishes vanishing, adjunction, and basepoint-freeness results for slc geometry in the analytic category. Collectively, these results extend the Mori program to highly singular complex analytic spaces and provide a unified framework to study slc and quasi-log phenomena analytically, with explicit bounds and constructive steps for generation and very ampleness.

Abstract

We introduce the notion of quasi-log complex analytic spaces and establish various fundamental properties. Moreover, we prove that a semi-log canonical pair naturally has a quasi-log complex analytic space structure. This paper is part of the author's project to establish a minimal model theory for projective morphisms between complex analytic spaces.
Paper Structure (13 sections, 46 theorems, 150 equations)

This paper contains 13 sections, 46 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On vanishing theorems
  4. Quasi-log structures for complex analytic spaces
  5. Some basic operations on quasi-log structures
  6. Basepoint-free theorem
  7. Basepoint-free theorem of Reid--Fukuda type
  8. Effective freeness
  9. Cone theorem
  10. On $\omega$-negative extremal rational curves
  11. On analytic semi-log canonical pairs
  12. Vanishing theorems and torsion-freeness for semi-log canonical pairs
  13. On Shokurov's polytopes for semi-log canonical pairs

Key Result

Theorem 1.3

Let be a quasi-log complex analytic space and let $X'$ be the union of $X_{-\infty}$ with a union of some qlc strata of $[X, \omega]$. Then, after replacing $X$ with any relatively compact open subset of $X$, we can construct a projective morphism $f'\colon (Y', B_{Y'})\to X'$ from an analytic globally is a quasi-log complex analytic space with $\omega'=\omega|_{X'}$ and $X'_{-\infty}=X_{-\infty

Theorems & Definitions (111)

  • Definition 1.1: Quasi-log complex analytic spaces, see Definition \ref{['a-def4.1']}
  • Remark 1.2: $\mathbb R$-line bundles and globally $\mathbb R$-Cartier divisors
  • Theorem 1.3: Adjunction, see Theorem \ref{['a-thm4.4']}
  • Theorem 1.4: Vanishing theorem, see Theorems \ref{['a-thm4.7']} and \ref{['a-thm4.8']}
  • Example 1.5: Normal pairs
  • Theorem 1.6: Effective freeness of Reid--Fukuda type for log canonical pairs
  • Theorem 1.7: Semi-log canonical pairs, see Theorem \ref{['a-thm10.1']}
  • Example 1.8
  • Remark 1.9: Minimal model program for projective morphisms of complex analytic spaces
  • Remark 1.10
  • ...and 101 more