Geometric quantization on big line bundles

TL;DR

This paper extends geometric quantization to big line bundles, establishing asymptotic isometry properties and linking filtrations to Mabuchi geodesic rays, with implications for complex geometry and algebraic structures.

Contribution

It introduces asymptotic isometry results for big line bundles and connects submultiplicative filtrations to Mabuchi geodesic rays, advancing geometric quantization theory.

Findings

Asymptotic isometry property for metrics on big line bundles.

Equivalence of submultiplicative norms and sup-norms asymptotically.

Filtrations induce Mabuchi geodesic rays encoding statistical invariants.

Abstract

We extend several geometric quantization results to the setting of big line bundles. More precisely, we prove the asymptotic isometry property for the map that associates to a metric on a big line bundle the corresponding sup-norms on the spaces of holomorphic sections of its tensor powers. Building on this, we show that submultiplicative norms on section rings of big line bundles are asymptotically equivalent to sup-norms. As an application, we show that any bounded submultiplicative filtration on the section ring of a big line bundle naturally gives rise to a Mabuchi geodesic ray, and the speed of this ray encodes the statistical invariants of the filtration.