On the Seshadri constants of equivariant bundles over Bott-Samelson varieties and wonderful compactifications

TL;DR

This paper investigates the positivity properties and Seshadri constants of torus-equivariant vector bundles on specific complex varieties, providing criteria for nefness and ampleness based on restrictions to invariant curves.

Contribution

It establishes a criterion linking nefness and ampleness of equivariant bundles to their restrictions on invariant curves, and computes Seshadri constants at fixed points.

Findings

Nefness and ampleness are characterized by restrictions to invariant curves.

Explicit computation of Seshadri constants at torus-fixed points.

Provides tools for understanding positivity of equivariant bundles on special varieties.

Abstract

We study torus-equivariant vector bundles $E$ on a complex projective variety $X$ which is either a Bott-Samelson-Demazure-Hansen variety or a wonderful compactification of a complex symmetric variety of minimal rank. We show that $E$ is nef (respectively, ample) if and only if its restriction to every torus--invariant curve in $X$ is nef (respectively, ample). We also compute the Seshadri constants $\varepsilon(E,x)$, where $x\, \in\, X$ is any point fixed by the action of a maximal torus.