Connections and genuinely ramified maps of curves

TL;DR

This paper investigates the structure of singular connections on curves, establishing conditions under which certain subsheaves induce nonsingular connections and characterizing genuinely ramified maps via fundamental group homomorphisms.

Contribution

It introduces the concept of maximal subsheaves inducing nonsingular connections and characterizes genuinely ramified maps through properties of these subsheaves and fundamental group surjectivity.

Findings

Unique maximal subsheaf with nonsingular connection exists for singular connections.

Surjectivity of étale fundamental group homomorphism characterized by maximal semistable subsheaf.

Genuinely ramified maps characterized by equality of a sheaf and its maximal subsheaf in characteristic zero.

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 $\phi : Y \rightarrow\ X$ 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 $V ={\bf R}(\phi_*\phi^*V)$.