Equivariant vector bundles on complete symmetric varieties of minimal rank

TL;DR

This paper characterizes the positivity and triviality of equivariant vector bundles on the wonderful compactification of minimal rank symmetric spaces by examining their restrictions to specific subvarieties.

Contribution

It establishes criteria for nefness, ampleness, and triviality of equivariant vector bundles on these compactifications based on their restrictions to certain subvarieties.

Findings

E is nef iff E|_Z is nef

E is ample iff E|_Z is ample

E is trivial iff E|_Z is trivial

Abstract

Let $X$ be the wonderful compactification of a complex symmetric space $G/H$ of minimal rank. For a point $x\,\in\, G$, denote by $Z$ be the closure of $BxH/H$ in $X$, where $B$ is a Borel subgroup of $G$. The universal cover of $G$ is denoted by $\widetilde{G}$. Given a $\widetilde{G}$ equivariant vector bundle $E$ on $X,$ we prove that $E$ is nef (respectively, ample) if and only if its restriction to $Z$ is nef (respectively, ample). Similarly, $E$ is trivial if and only if its restriction to $Z$ is so.