Submultiplicative norms and filtrations on section rings

TL;DR

This paper establishes the asymptotic equivalence between submultiplicative norms and sup-norms on section rings of polarized projective manifolds, with applications to spectral theory, pseudonorms, and tensor norms.

Contribution

It demonstrates the asymptotic equivalence of submultiplicative norms and sup-norms, and explores implications for spectral theory, pseudonorms, and tensor norms.

Findings

Submultiplicative norms are asymptotically equivalent to sup-norms.

Injective and projective tensor norms are asymptotically equivalent.

Applications to spectral theory and holomorphic extension theorems.

Abstract

We show that submultiplicative norms on section rings of polarised projective manifolds are asymptotically equivalent to sup-norms associated with metrics on the polarisation. We then discuss some applications to the spectral theory of submultiplicative filtrations, the asymptotic study of the Narasimhan-Simha pseudonorms, and holomorphic extension theorem. As an unexpected byproduct, we show that injective and projective tensor norms on symmetric algebras of finite dimensional complex normed vector spaces are asymptotically equivalent.