Kodaira-Iitaka dimension and multiplicity: an analytic perspective

TL;DR

This paper provides an analytic framework connecting the Kodaira-Iitaka dimension and multiplicity of linear series with intersection theory and plurisubharmonic envelopes, introducing refined formulas and a new invariant.

Contribution

It introduces a novel analytic approach to express Kodaira-Iitaka dimension and multiplicity via intersection theory and plurisubharmonic envelopes, including refined pointwise and metric formulas.

Findings

Pointwise convergence of Bergman kernels analyzed

Asymptotic volume ratios computed

A non-pluripolar numerical Kodaira-Iitaka dimension introduced

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.