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Metrics with minimal singularities and the Abundance conjecture

Vladimir Lazić

Abstract

The Abundance conjecture predicts that on a minimal projective klt pair $(X,Δ)$, the adjoint divisor $K_X+Δ$ is semiample. When $χ(X,\mathcal O_X)\neq0$, we give a necessary and sufficient condition for the conjecture to hold in terms of the asymptotic behaviour of multiplier ideals of currents with minimal singularities of small twists of $K_X+Δ$. Furthermore, we prove fundamental structural properties as well as regularity and weak convergence behaviour of an important class of currents with minimal singularities: the supercanonical currents. The results of the paper indicate strongly that supercanonical currents are central to the completion of the proof of the Abundance conjecture for minimal klt pairs $(X,Δ)$ with $χ(X,\mathcal O_X)\neq0$.

Metrics with minimal singularities and the Abundance conjecture

Abstract

The Abundance conjecture predicts that on a minimal projective klt pair , the adjoint divisor is semiample. When , we give a necessary and sufficient condition for the conjecture to hold in terms of the asymptotic behaviour of multiplier ideals of currents with minimal singularities of small twists of . Furthermore, we prove fundamental structural properties as well as regularity and weak convergence behaviour of an important class of currents with minimal singularities: the supercanonical currents. The results of the paper indicate strongly that supercanonical currents are central to the completion of the proof of the Abundance conjecture for minimal klt pairs with .

Paper Structure

This paper contains 48 sections, 46 theorems, 340 equations.

Table of Contents

  1. Introduction
  2. Notation.
  3. On the organisation of the paper.
  4. Acknowledgements.
  5. Preliminaries
  6. Preliminaries: pluripotential theory
  7. Bott-Chern cohomology
  8. Almost positive currents
  9. Plurisubharmonic functions
  10. Quasi-psh functions
  11. Upper semicontinuous regularisation
  12. Positivity of classes
  13. Lelong numbers
  14. Divisorial valuations
  15. Siu decomposition
  16. ...and 33 more sections

Key Result

Proposition 1.1

Let $(X,\Delta)$ be a projective klt pair such that $K_X+\Delta$ is semiample. Let $\pi\colon Y\to X$ be a log resolution of $(X,\Delta)$ and write where $\Delta_Y$ and $E$ are effective $\mathbb{R}$-divisors without common components. Let $A$ be an ample $\mathbb{R}$-divisor on $Y$. Then there exist an effective divisor $D$ on $Y$ and a sequence of positive integers $\{m_\ell\}_{\ell\in\mathbb{N

Theorems & Definitions (104)

  • Proposition 1.1
  • Theorem 1
  • Theorem 2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Theorem 2.4
  • ...and 94 more