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Rigid currents in birational geometry

Vladimir Lazić, Zhixin Xie

Abstract

A rigid current on a compact complex manifold is a closed positive current whose cohomology class contains only one closed positive current. Rigid currents occur in complex dynamics, algebraic and differential geometry. The goals of the present paper are: (a) to give a systematic treatment of rigid currents, (b) to demonstrate how they appear within the Minimal Model Program, and (c) to give many new examples of rigid currents.

Rigid currents in birational geometry

Abstract

A rigid current on a compact complex manifold is a closed positive current whose cohomology class contains only one closed positive current. Rigid currents occur in complex dynamics, algebraic and differential geometry. The goals of the present paper are: (a) to give a systematic treatment of rigid currents, (b) to demonstrate how they appear within the Minimal Model Program, and (c) to give many new examples of rigid currents.
Paper Structure (14 sections, 13 theorems, 62 equations)

This paper contains 14 sections, 13 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pairs
  4. Minimal and good models
  5. Invariant Iitaka dimension
  6. Positive currents
  7. Plurisubharmonic functions
  8. Lelong numbers
  9. Nakayama--Zariski functions and Boucksom--Zariski functions
  10. Rigid currents
  11. Rigid currents and fibrations
  12. Rigidity and currents with minimal singularities
  13. Weak rigidity
  14. Proofs of the main results

Key Result

Theorem 1.1

Assume the existence of good models for projective log canonical pairs in dimension $n-1$. Let $(X,\Delta)$ be a projective klt pair of dimension $n$ such that $\kappa_\iota(X,K_X+\Delta )=0$, and let $D\geq0$ be the unique $\mathbb{R}$-divisor such that $K_X+\Delta\sim_\mathbb{R} D$. Let $f\colon Y

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Remark 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • ...and 28 more