Connections and genuinely ramified maps of curves

Abstract

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 $\varphi :Y\to X$ between irreducible smooth projective curves, and a singular connection $(V,D)$ on $Y$, the direct image ${\varphi }_{\ast }V$ has a singular connection. Let $\text{R}({\varphi }_{\ast }{\mathcal{O}}_Y)$ be the unique maximal subsheaf on which the singular connection on ${\varphi }_{\ast }{\mathcal{O}}_Y$ -- corresponding to the trivial connection on ${\mathcal{O}}_Y$ -- induces a nonsingular connection. We prove that the homomorphism of étale fundamental groups ${\varphi }_{\ast }:{\pi }_1^{\mathrm{et}}(Y,y_0)\to {\pi }_1^{\mathrm{et}}(X,\varphi (y_0))$ induced by $\varphi$ is surjective if and only if ${\mathcal{O}}_X\subset \text{R}({\varphi }_{\ast }{\mathcal{O}}_Y)$ is the unique maximal semistable subsheaf. When the characteristic of the base field is zero, this homomorphism ${\varphi }_{\ast }$ is surjective if and only if ${\mathcal{O}}_X=\text{R}({\varphi }_{\ast }{\mathcal{O}}_Y)$. For any nonsingular connection $D$ on a vector bundle $V$ over $X$, there is a natural map $V\hookrightarrow \mathbf{R}({\varphi }_{\ast }{\varphi }^{\ast }V)$. When the characteristic of the base field is zero, we prove that the map $\varphi$ is genuinely ramified if and only if $V=\mathbf{R}({\varphi }_{\ast }{\varphi }^{\ast }V)$.

Final version; Forum Mathematicum (to appear),