On the direct images of parabolic vector bundles and parabolic connections

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 $V_<em>$ be a parabolic vector bundle on $Y$. We construct a parabolic structure on the direct image $\varphi_</em> V$ on $X$, where $V$ is the vector bundle underlying $V_<em>$. The parabolic vector bundle $\varphi_</em> V_<em>$ on $X$ obtained this way has a ramified torus sub-bundle; it is a torus bundle of $\text{Ad}(\varphi_</em> V)$ outside the parabolic divisor for $\varphi_* V_<em>$ that satisfies certain conditions at the parabolic points. Conversely, given a parabolic vector bundle $E_</em>$ 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 $W_<em>$ on $Z$, such that the parabolic bundle $E_</em>$ is the direct image of $W_<em>$. A connection on $V_</em>$ produces a connection on $\varphi_* V_<em>$. The ramified torus sub-bundle for $\varphi_</em> V_<em>$ is preserved by the logarithmic connection on $\text{End}(\varphi_</em> V)$ induced by this connection on $\varphi_* V_<em>$. If the parabolic vector bundle $E_</em>$ 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 $W_<em>$ on $Z$ has a connection that produces the connection $D$ on the direct image $E_</em>$.