On the direct images of parabolic vector bundles and parabolic connections

TL;DR

This paper studies how parabolic vector bundles and connections behave under direct images via finite morphisms between curves, establishing constructions and conditions for preserving structures like torus sub-bundles and connections.

Contribution

It introduces a method to construct parabolic structures on direct images and characterizes when connections preserve these structures, linking bundles on different curves.

Findings

Constructed parabolic structures on direct images of bundles.

Characterized conditions for preserving torus sub-bundles under connections.

Established correspondence between connections on bundles and their direct images.

Abstract

Let $\varphi : Y \rightarrow X$ be a finite surjective morphism between smooth complex projective curves, where $X$ is irreducible but $Y$ need not be so. Let $V_*$ be a parabolic vector bundle on $Y$. We construct a parabolic structure on the direct image $\varphi_* V$ on $X$, where $V$ is the vector bundle underlying $V_*$. The parabolic vector bundle $\varphi_* V_*$ on $X$ obtained this way has a ramified torus sub-bundle; it is a torus bundle of $\text{Ad}(\varphi_* V)$ outside the parabolic divisor for $\varphi_* V_*$ that satisfies certain conditions at the parabolic points. Conversely, given a parabolic vector bundle $E_*$ on $X$, and a ramified torus sub-bundle $\mathcal T$ for it, we construct a ramified covering $Z$ of $X$ and a parabolic vector bundle $W_*$ on $Z$, such that the parabolic bundle $E_*$ is the direct image of $W_*$. A connection on $V_*$ produces a connection on $\varphi_* V_*$. The ramified torus sub-bundle for $\varphi_* V_*$ is preserved by the logarithmic connection on $\text{End}(\varphi_* V)$ induced by this connection on $\varphi_* V_*$. If the parabolic vector bundle $E_*$ on $X$ is equipped with a connection $D$ such that the connection on the endomorphism bundle induced by it preserves the ramified torus sub-bundle $\mathcal T$, then we prove that the corresponding parabolic vector bundle $W_*$ on $Z$ has a connection that produces the connection $D$ on the direct image $E_*$.