b-divisorial valuations and Berkovich positivity functions

TL;DR

This paper extends positivity invariants like Seshadri constants to Berkovich spaces, proving their semicontinuity and connecting them to b-divisors and valuations.

Contribution

It introduces a framework for analyzing local positivity invariants via b-divisors and seminorms in Berkovich spaces, extending classical concepts.

Findings

Semicontinuity of positivity invariants as functions of center seminorms.

Extension of Seshadri constants to all seminorms in Berkovich space.

Use of b-divisors to translate positivity questions into cone shape analysis.

Abstract

We prove semicontinuity properties for local positivity invariants of big and nef divisors. The usual definition of Seshadri constant and asymptotic order of vanishing along a subvariety is extended to include all seminorms in the Berkovich space, and we obtain semicontinuity of such constants as a function of the center seminorm. We use Shokurov's language of b-divisors; to each seminorm there is an associated b-divisor which can be used to translate questions about positivity into questions about the shape of certain cones of b-divisors. The theory works especially well for what we call b-divisorial valuations, a natural extension of the notion of divisorial valuations which encompasses, e.g., all Abhyankar valuations.