Connections and genuinely ramified maps of curves

Given a singular connection $D$ on a vector bundle $E$ over an irreducible smooth projective curve $X$, defined over an algebraically closed field, we show that there is a unique maximal subsheaf of $E$ on which $D$ induces a nonsingular connection. Given a generically smooth map $\phi : Y \rightarrow\ X$ between irreducible smooth projective curves, and a singular connection $(V, D)$ on $Y$, the direct image $\phi_*V$ has a singular connection. Let $\textbf{R}(\phi_*{\mathcal O}_Y)$ be the unique maximal subsheaf on which the singular connection on $\phi_*{\mathcal O}_Y$ -- corresponding to the trivial connection on ${\mathcal O}_Y$ -- induces a nonsingular connection. We prove that the homomorphism of \'etale fundamental groups $\phi_*: \pi_1^{\rm et}(Y, y_0) \rightarrow \pi_1^{\rm et}(X, \phi(y_0))$ induced by $\phi$ is surjective if and only if ${\mathcal O}_X \subset \textbf{R}(\phi_*{\mathcal O}_Y)$ is the unique maximal semistable subsheaf. When the characteristic of the base field is zero, this homomorphism $\phi_*$ is surjective if and only if ${\mathcal O}_X = \textbf{R}(\phi_*{\mathcal O}_Y)$. For any nonsingular connection $D$ on a vector bundle $V$ over $X$, there is a natural map $V\hookrightarrow {\bf R}(\phi_*\phi^*V)$. When the characteristic of the base field is zero, we prove that the map $\phi$ is genuinely ramified if and only if $V ={\bf R}(\phi_*\phi^*V)$.