Genuinely ramified maps and pseudo-stable vector bundles

Abstract

Let X and Y be irreducible normal projective varieties, of same dimension, defined over an algebraically closed field, and let $f:Y⟶X$ be a finite generically smooth morphism such that the corresponding homomorphism between the étale fundamental groups $f_{\ast }:{\mathrm{\pi }}_1^{\mathrm{et}}(Y)⟶{\mathrm{\pi }}_1^{\mathrm{et}}(X)$ is surjective. Fix a polarization on X and equip Y with the pulled-back polarization. For a point $y_0\in Y$, let $\mathrm{\varpi }(Y,y_0)$ (resp. $\mathrm{\varpi }(X,f(y_0))$) be the affine group scheme given by the neutral Tannakian category defined by the strongly pseudo-stable vector bundles of degree zero on Y (resp. X). We prove that the homomorphism $\mathrm{\varpi }(Y,y_0)⟶\mathrm{\varpi }(X,f(y_0))$ induced by f is surjective. Let E be a pseudo-stable vector bundle on X and $F\subset f^{\ast }E$ a pseudo-stable subbundle with $\mathrm{\mu }(F)=\mathrm{\mu }(f^{\ast }E)$. We prove that $f^{\ast }E$ is pseudo-stable and there is a pseudo-stable subbundle $W\subset E$ such that $f^{\ast }W=F$ as subbundles of $f^{\ast }E$.