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Transcendental b-divisors I -- Correspondence with currents

Mingchen Xia

Abstract

We study transcendental b-divisors over compact Kähler manifolds. We establish the correspondence between closed positive currents and nef b-divisors. As an application, we establish the intersection theory of nef b-divisors, answering a question of Dang--Favre.

Transcendental b-divisors I -- Correspondence with currents

Abstract

We study transcendental b-divisors over compact Kähler manifolds. We establish the correspondence between closed positive currents and nef b-divisors. As an application, we establish the intersection theory of nef b-divisors, answering a question of Dang--Favre.
Paper Structure (16 sections, 53 theorems, 168 equations)

This paper contains 16 sections, 53 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modifications and cones
  4. Quasi-plurisubharmonic functions
  5. Mixed volumes
  6. The different definitions
  7. Properties of mixed volumes
  8. Transcendental b-divisors
  9. The definitions
  10. The b-divisors of currents
  11. The intersection theory
  12. Smooth pull-backs of b-divisors
  13. The trace operator of b-divisors
  14. The analytic theory
  15. The codimension 1 case
  16. ...and 1 more sections

Key Result

Theorem 1.1

There is a natural map $\mathbb{D}$ (via Siu's decomposition) sending each closed positive $(1,1)$-current $T$ on $X$ to a nef b-divisor $\mathbb{D}(T)$ over $X$. Let $\alpha$ be a modified nef cohomology class on $X$ and $\mathbb{D}$ be a nef and big b-divisor over $X$ with $\mathbb{D}_X=\alpha$, t

Theorems & Definitions (116)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3: \ref{['cor:nef_Cartierseq']}
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • ...and 106 more