Genuinely ramified maps and pseudo-stable vector bundles

TL;DR

This paper investigates the properties of pseudo-stable vector bundles under genuinely ramified maps between algebraic varieties, establishing surjectivity of certain fundamental group homomorphisms and stability preservation.

Contribution

It proves the surjectivity of the induced homomorphism between affine group schemes and shows that pseudo-stability is preserved under pullback and subbundle conditions.

Findings

Surjectivity of the homomorphism between affine group schemes induced by the map.

Pseudo-stability of the pullback of a pseudo-stable vector bundle.

Existence of a pseudo-stable subbundle W such that f^*W = F.

Abstract

Let $X$ and $Y$ be irreducible normal projective varieties, of same dimension, defined over an algebraically closed field, and let $f : Y \rightarrow X$ be a finite generically smooth morphism such that the corresponding homomorphism between the \'etale fundamental groups $f_*:\pi^{\rm et}_{1}(Y) \rightarrow\pi^{\rm et}_{1}(X)$ is surjective. Fix a polarization on $X$ and equip $Y$ with the pulled back polarization. For a point $y_0\in Y$, let $\varpi(Y, y_0)$ (respectively, $\varpi(X, f(y_0))$) be the affine group scheme given by the neutral Tannakian category defined by the strongly pseudo-stable vector bundles of degree zero on $Y$ (respectively, $X$). We prove that the homomorphism $\varpi(Y, y_0) \rightarrow \varpi(X, f(y_0))$ induced by $f$ is surjective. Let $E$ be a pseudo-stable vector bundle on $X$ and $F \subset f^*E$ a pseudo-stable subbundle with $\mu(F)= \mu(f^*E)$. We prove that $f^*E$ is pseudo-stable and there is a pseudo-stable subbundle $W \subset E$ such that $f^*W = F$ as subbundles of $f^*E$.