Canonical growth conditions associated to ample line bundles

TL;DR

This paper introduces a canonical growth condition associated with ample line bundles on projective manifolds, encoding classical invariants and infinitesimal Okounkov bodies, with implications for Kähler geometry and embeddings.

Contribution

It constructs a new growth condition that captures key invariants and infinitesimal structures of line bundles, extending toric geometry concepts to broader settings.

Findings

Encodes volume and Seshadri constants via growth conditions

Recovers all infinitesimal Okounkov bodies at a point

Generalizes Gromov width results for Kähler embeddings

Abstract

We propose a new construction which associates to any ample (or big) line bundle $L$ on a projective manifold $X$ a canonical growth condition (i.e. a choice of a psh function well-defined up to a bounded term) on the tangent space $T_p X$ of any given point $p$. We prove that it encodes such classical invariants as the volume and the Seshadri constant. Even stronger, it allows you to recover all the infinitesimal Okounkov bodies of $L$ at $p$. The construction is inspired by toric geometry and the theory of Okounkov bodies; in the toric case the growth condition is "equivalent" to the moment polytope. As in the toric case the growth condition says a lot about the K\"ahler geometry of the manifold. We prove a theorem about K\"ahler embeddings of large balls, which generalizes the well-known connection between Seshadri constants and Gromov width established by McDuff and Polterovich.