Projective bundles over toric surfaces

TL;DR

This paper investigates how the cohomology ring of non-singular projective toric varieties determines their structure as projective bundles over toric surfaces, and classifies certain higher-dimensional bundles over quasitoric manifolds.

Contribution

It establishes the relationship between cohomology rings and bundle structures over toric varieties and quasitoric manifolds, providing classification results.

Findings

Cohomology ring determines projective bundle structure over toric surfaces.

Two 6-dimensional bundles over 4-dimensional quasitoric manifolds are diffeomorphic if their cohomology rings are isomorphic.

Studied smooth classification of higher-dimensional bundles over quasitoric manifolds.

Abstract

Let $E$ be the Whitney sum of complex line bundles over a topological space $X$. Then, the projectivization $P(E)$ of $E$ is called a \emph{projective bundle} over $X$. If $X$ is a non-singular complete toric variety, so is $P(E)$. In this paper, we show that the cohomology ring of a non-singular projective toric variety $M$ determines whether it admits a projective bundle structure over a non-singular complete toric surface. In addition, we show that two 6-dimensional projective bundles over 4-dimensional quasitoric manifolds are diffeomorphic if their cohomology rings are isomorphic as graded rings. Furthermore, we study the smooth classification of higher dimensional projective bundles over 4-dimensional quasitoric manifolds.