Connections and genuinely ramified maps of curves

Abstract

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 ϕ : Y → X {\phi:Y\rightarrow X} between irreducible smooth projective curves, and a singular connection ( V , D ) {(V,D)} on Y, the direct image ϕ * ⁢ V {\phi_{*}V} has a singular connection. Let 𝐑 ⁢ ( ϕ * ⁢ 𝒪 Y ) {\mathbf{R}(\phi_{*}{\mathcal{O}}_{Y})} be the unique maximal subsheaf on which the singular connection on ϕ * ⁢ 𝒪 Y {\phi_{*}{\mathcal{O}}_{Y}} – corresponding to the trivial connection on 𝒪 Y {{\mathcal{O}}_{Y}} – induces a nonsingular connection. We prove that the homomorphism of étale fundamental groups ϕ * : π 1 et ⁢ ( Y , y 0 ) → π 1 et ⁢ ( X , ϕ ⁢ ( y 0 ) ) {\phi_{*}:\pi_{1}^{\rm et}(Y,y_{0})\rightarrow\pi_{1}^{\rm et}(X,\phi(y_{0}))} induced by ϕ is surjective if and only if 𝒪 X ⊂ 𝐑 ⁢ ( ϕ * ⁢ 𝒪 Y ) {{\mathcal{O}}_{X}\subset\mathbf{R}(\phi_{*}{\mathcal{O}}_{Y})} is the unique maximal semistable subsheaf. When the characteristic of the base field is zero, this homomorphism ϕ * {\phi_{*}} is surjective if and only if 𝒪 X = 𝐑 ⁢ ( ϕ * ⁢ 𝒪 Y ) {{\mathcal{O}}_{X}=\mathbf{R}(\phi_{*}{\mathcal{O}}_{Y})} . For any nonsingular connection D on a vector bundle V over X, there is a natural map V ↪ 𝐑 ⁢ ( ϕ * ⁢ ϕ * ⁢ V ) {V\hookrightarrow\mathbf{R}(\phi_{*}\phi^{*}V)} . When the characteristic of the base field is zero, we prove that the map ϕ is genuinely ramified if and only if V = 𝐑 ⁢ ( ϕ * ⁢ ϕ * ⁢ V ) {V=\mathbf{R}(\phi_{*}\phi^{*}V)} .