Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Kodaira-Iitaka dimension and multiplicity: an analytic perspective

Siarhei Finski

Abstract

We express the Kodaira-Iitaka dimension and the multiplicity of graded linear series in terms of the intersection theory of the plurisubharmonic envelope associated with the linear series, and obtain two refined versions of these formulas at the pointwise and at the metric levels. At the pointwise level, we focus on the weak convergence of the partial Bergman kernel associated with the linear series and a Bernstein-Markov measure. At the metric level, we compute the asymptotic ratio of the volumes of unit balls defined by the sup-norms on the linear series. Based on our findings, we introduce a non-pluripolar version of the numerical Kodaira-Iitaka dimension for a line bundle, show that this invariant dominates the classical Kodaira-Iitaka dimension and is, in turn, bounded above by the numerical versions proposed so far.

Kodaira-Iitaka dimension and multiplicity: an analytic perspective

Abstract

We express the Kodaira-Iitaka dimension and the multiplicity of graded linear series in terms of the intersection theory of the plurisubharmonic envelope associated with the linear series, and obtain two refined versions of these formulas at the pointwise and at the metric levels. At the pointwise level, we focus on the weak convergence of the partial Bergman kernel associated with the linear series and a Bernstein-Markov measure. At the metric level, we compute the asymptotic ratio of the volumes of unit balls defined by the sup-norms on the linear series. Based on our findings, we introduce a non-pluripolar version of the numerical Kodaira-Iitaka dimension for a line bundle, show that this invariant dominates the classical Kodaira-Iitaka dimension and is, in turn, bounded above by the numerical versions proposed so far.
Paper Structure (13 sections, 27 theorems, 178 equations)

This paper contains 13 sections, 27 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Linear series: a case study
  3. Analytic formulas for the Kodaira-Iitaka dimension and the multiplicity
  4. Preliminaries from pluripotential theory
  5. Fujita-type approximation theorem and resolutions
  6. Fubini-Study maps and intersection theory
  7. Comparison of the volumes of unit balls
  8. An alternative quantization scheme for the envelope
  9. Ratio of volumes and the relative Monge-Ampère $\kappa$-energy
  10. Asymptotics of the partial Bergman kernels
  11. An example of divergent partial Bergman kernels
  12. Partial differentiability of the relative Monge-Ampère $\kappa$-energy
  13. Integral closure of linear series and singularity types

Key Result

Theorem 1.1

For any graded linear series $W \subset R(X, L)$, we have

Theorems & Definitions (58)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 48 more