On the direct images of parabolic vector bundles and parabolic connections

Abstract

Let φ:Y⟶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 φ∗V on X, where V is the vector bundle underlying V∗. The parabolic vector bundle φ∗V∗ on X obtained this way has a ramified torus sub-bundle; it is a torus bundle of Ad(φ∗V) outside the parabolic divisor for φ∗V∗ that satisfies certain conditions at the parabolic points. Conversely, given a parabolic vector bundle E∗ on X, and a ramified torus sub-bundle 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 φ∗V∗. The ramified torus sub-bundle for φ∗V∗ is preserved by the logarithmic connection on End(φ∗V) induced by this connection on φ∗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 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∗.