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 $ϕ: Y \rightarrow\ X$ between irreducible smooth projective curves, and a singular connection $(V, D)$ on $Y$, the direct image $ϕ_*V$ has a singular connection. Let $\textbf{R}(ϕ_*{\mathcal O}_Y)$ be the unique maximal subsheaf on which the singular connection on $ϕ_*{\mathcal O}_Y$ -- corresponding to the trivial connection on ${\mathcal O}_Y$ -- induces a nonsingular connection. We prove that the homomorphism of étale fundamental groups $ϕ_*: π_1^{\rm et}(Y, y_0) \rightarrow π_1^{\rm et}(X, ϕ(y_0))$ induced by $ϕ$ is surjective if and only if ${\mathcal O}_X \subset \textbf{R}(ϕ_*{\mathcal O}_Y)$ is the unique maximal semistable subsheaf. When the characteristic of the base field is zero, this homomorphism $ϕ_*$ is surjective if and only if ${\mathcal O}_X = \textbf{R}(ϕ_*{\mathcal O}_Y)$. For any nonsingular connection $D$ on a vector bundle $V$ over $X$, there is a natural map $V\hookrightarrow {\bf R}(ϕ_*ϕ^*V)$. When the characteristic of the base field is zero, we prove that the map $ϕ$ is genuinely ramified if and only if $V ={\bf R}(ϕ_*ϕ^*V)$.

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