Pseudo-effective cones of projective bundles and weak Zariski decomposition

TL;DR

This paper characterizes when the nef and pseudoeffective cones of divisors coincide on projective bundles over complex varieties, establishing conditions for weak Zariski decompositions and cone equalities in specific cases.

Contribution

It provides necessary and sufficient conditions for cone equality on projective bundles with semistable bundles and demonstrates the existence of weak Zariski decompositions in these contexts.

Findings

Nef and pseudoeffective cones coincide under certain conditions.

Weak Zariski decomposition exists on specific projective bundles.

Semistable bundles with zero second Chern class are k-homogeneous.

Abstract

In this article, we consider the projective bundle $\mathbb{P}_X(E)$ over a smooth complex projective variety $X$, where $E$ is a semistable bundle on $X$ with $c_2(End(E)) =0$. We give a necessary and sufficient condition to get the equality $ Nef^1\bigl(\mathbb{P}_X(E)\bigr) = \overline{Eff}^1\bigl(\mathbb{P}_X(E)\bigr)$ of nef cone and pseudoeffective cone of divisors in $\mathbb{P}_X(E)$. As an application of our result, we show the equality of nef and pseudoeffective cones of divisors of projective bundles over some special varieties. In particular, we show that weak Zariski decomposition exists on these projective bundles. We also show that a semistable bundle $E$ of rank $r \geq 2$ with $c_2\bigl(End(E)\bigr) = 0$ on a smooth complex projective variety of Picard number 1 is $k$-homogeneous i.e. $\overline{Eff}^k\bigl(\mathbb{P}_X(E)\bigr) = Nef^k\bigl(\mathbb{P}_{X}(E)\bigr)$ for all $1 \leq k < r$. Finally, we show that weak Zariski decomposition exists for a fibre product $\mathbb{P}_C(E)\times_C\mathbb{P}(E')$ over a smooth projective curve $C$.