# Successive minima of line bundles

## TL;DR

This paper introduces the concept of successive minima for line bundles on algebraic varieties, linking geometric invariants like volume, width, and Seshadri constants, and relates them to toric geometry.

## Contribution

It defines and studies successive minima of line bundles, connecting them to classical invariants and extending their understanding in algebraic and toric geometry.

## Key findings

- Successive minima relate to width and Seshadri constants.
- Volume equals the product of successive minima.
- In toric varieties, minima correspond to classical lattice minima.

## Abstract

We introduce and study the successive minima of line bundles on proper algebraic varieties. The first (resp. last) minima are the width (resp. Seshadri constant) of the line bundle at very general points. The volume of the line bundle is equivalent to the product of the successive minima. For line bundles on toric varieties, the successive minima are equivalent to the (reciprocal of) usual successive minima of the difference of the moment polytope.