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Transcendental b-divisors II -- Monotonicity theorem

Mingchen Xia

Abstract

In this paper, we develop the general intersection theory of nef b-divisors, extending the movable intersection theory. We define a notion of restricted volume of b-divisors and prove a quantitative version of the monotonicity of the intersection product. As a consequence, we prove a number of new volume inequalities of currents and cohomology classes.

Transcendental b-divisors II -- Monotonicity theorem

Abstract

In this paper, we develop the general intersection theory of nef b-divisors, extending the movable intersection theory. We define a notion of restricted volume of b-divisors and prove a quantitative version of the monotonicity of the intersection product. As a consequence, we prove a number of new volume inequalities of currents and cohomology classes.
Paper Structure (24 sections, 83 theorems, 300 equations, 2 figures, 2 tables)

This paper contains 24 sections, 83 theorems, 300 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Acknowledgments
  5. Preliminaries
  6. Quasi-plurisubharmonic functions
  7. Modifications and cones
  8. Partially ordered linear spaces
  9. Cones in cohomology
  10. Movable intersection theory
  11. Generalized intersection theory
  12. The notion of b-classes
  13. The intersection theory
  14. Restricted volume
  15. The trace operator
  16. ...and 9 more sections

Key Result

Theorem A

Figures (2)

  • Figure 1: The optimal situation
  • Figure 2: The Okounkov bodies

Theorems & Definitions (184)

  • Theorem A
  • Theorem B
  • Corollary 1.1
  • Theorem 1.2
  • Theorem C
  • Theorem 1.3
  • Theorem D
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1
  • ...and 174 more