Canonical subsheaves of torsionfree semistable sheaves - Biswas, Indranil; Parameswaran, A. J. (MATHEMATICA SCANDINAVICA, 2022)
Let $F$ be a torsionfree semistable coherent sheaf on a polarized normal projective variety defined over an algebraically closed field. We prove that $F$ has a unique maximal locally free subsheaf $V$ such that $F/V$ is torsionfree and $V$ also admits a filtration of subbundles for which each successive quotient is a stable vector bundle whose slope is $μ (F)$. We also prove that $F$ has a unique maximal reflexive subsheaf $W$ such that $F/W$ is torsionfree and $W$ admits a filtration of subsheaves for which each successive quotient is a stable reflexive sheaf whose slope is $μ (F)$. We show that these canonical subsheaves behave well with respect to the pullback operation by étale Galois covering maps. Given a separable finite surjective map $φ \colon Y \longrightarrow X$ between normal projective varieties, we give a criterion for the induced homomorphism of étale fundamental groups $φ _*\colon π ^{et}_{1}(Y) \longrightarrow π ^{et}_{1}(X)$ to be surjective. The criterion in question is expressed in terms of the above mentioned unique maximal locally free subsheaf associated to the direct image $φ _*{\mathcal O}_Y$.