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An approach to the abundance conjecture for Kähler varieties via algebraic reduction

Zhiyuan Jiang

Abstract

In this article, we establish a strategy to the abundance conjecture for Kähler varieties via induction on algebraic dimension. Our strategy is to reduce the abundance conjecture for Kähler varieties to the abundance conjecture for projective varieties using the algebraic reduction fibration. In dimension 4, we apply our inductive strategy to obtain some cases of the abundance conjecture for Kähler fourfolds that are not algebraic or have trivial $K_X$.

An approach to the abundance conjecture for Kähler varieties via algebraic reduction

Abstract

In this article, we establish a strategy to the abundance conjecture for Kähler varieties via induction on algebraic dimension. Our strategy is to reduce the abundance conjecture for Kähler varieties to the abundance conjecture for projective varieties using the algebraic reduction fibration. In dimension 4, we apply our inductive strategy to obtain some cases of the abundance conjecture for Kähler fourfolds that are not algebraic or have trivial .

Paper Structure

This paper contains 16 sections, 41 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Analytic Geometry
  4. Positivity in Kähler Geometry
  5. Algebraic Reduction
  6. Zariski Decomposition
  7. b-divisors
  8. Zariski Decomposition
  9. Semi-stable Reduction in Codimension 1
  10. Branched Covering
  11. Extension of Ideal Sheaves
  12. Semi-stable Reduction in codimension one
  13. The Canonical Bundle Formula
  14. klt-trivial Fibrations
  15. Proof of the canonical bundle formula
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(X,\Delta)$ be a Kähler klt log pair. Assume If $K_X+\Delta$ is nef, then $K_X+\Delta$ is semiample. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (97)

  • Conjecture 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Flattening Theorem
  • Remark 2.4
  • Proposition 2.5
  • ...and 87 more