Direct image of parabolic line bundles

TL;DR

This paper establishes criteria for when a vector bundle on a projective variety is a direct image of a line bundle under an étale morphism, involving Cartan subalgebra bundles, with applications to parabolic bundles.

Contribution

It provides necessary and sufficient conditions for vector bundles to be direct images of line bundles under étale morphisms, including parabolic structures, advancing understanding of bundle origins.

Findings

Criterion based on Cartan subalgebra bundles for direct image of line bundles.

Characterization of when a bundle is a direct image of the structure sheaf.

Extension of criteria to parabolic vector bundles under ramified covers.

Abstract

Given a vector bundle $E$ on an irreducible projective variety $X$ we give a necessary and sufficient criterion for $E$ to be a direct image of a line bundle under an \'etale morphism. The criterion in question is the existence of a Cartan subalgebra bundle of the endomorphism bundle $\text{End}(E)$. As a corollary, a criterion is obtained for $E$ to be the direct image of the structure sheaf under an \'etale morphism. The direct image of a parabolic line bundle under any ramified covering map has a natural parabolic structure. Given a parabolic vector bundle, we give a similar criterion for it to be a direct image of a parabolic line bundle under a ramified covering map.