Slope Semistability and Positive cones of Grassmann bundles

TL;DR

This paper characterizes the nef and pseudoeffective cones of divisors in Grassmann bundles over complex varieties, linking their structure to the slope semistability of the underlying vector bundle.

Contribution

It provides explicit descriptions of these cones and establishes a criterion connecting their equality to the slope semistability and vanishing second Chern class of the vector bundle.

Findings

Nef and pseudoeffective cones coincide iff E is slope semistable with c2(End(E))=0.

Explicit computation of cones for Grassmann bundles.

Conditions for nefness and ampleness of the universal quotient bundle.

Abstract

Let $E$ be a vector bundle of rank $r$ on a smooth complex projective variety $X$. In this article, we compute the nef and pseudoeffective cones of divisors in the Grassmann bundle $Gr_X(k,E)$ parametrizing $k$-dimensional subspaces of the fibers of $E$, where $1\leq k \leq rank(E)$, under assumptions on $X$ as well as on the vector bundle $E$. In particular, we show that nef cone and the pseudoeffective cone of $Gr_X(k,E)$ coincide if and only if $E$ is a slope semistable bundle on $X$ with $c_2(End(E))=0$. We also discuss about the nefness and ampleness of the universal quotient bundle $Q_k$ on $Gr_X(k,E)$.