Canonical subsheaves of torsionfree semistable sheaves

TL;DR

This paper establishes the existence and uniqueness of canonical subsheaves within torsionfree semistable sheaves on normal projective varieties, and explores their behavior under étale covers and finite maps.

Contribution

It introduces canonical maximal locally free and reflexive subsheaves for torsionfree semistable sheaves, and analyzes their stability, slopes, and behavior under pullbacks and finite covers.

Findings

Existence and uniqueness of maximal locally free subsheaves.

Existence and uniqueness of maximal reflexive subsheaves.

Behavior of these subsheaves under étale Galois coverings.

Abstract

Let $F$ be a torsionfree semistable coherent sheaf on a polarized normal projective variety. We prove that $F$ has a unique maximal locally free subsheaf $V$ such that $F/V$ is torsionfree and $V$ admits a filtration of subbundles for which each successive quotient is stable whose slope is $\mu(F)$. 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 $\mu(F)$. We show that these canonical subsheaves behave well with respect to the pullback operation by \'etale Galois covering maps. Given a separable finite surjective map $\phi :Y \, \rightarrow X$ between normal projective varieties, we give a criterion for the induced homomorphism of \'etale fundamental groups $\phi_* : \pi^{\rm et}_{1}(Y) \rightarrow \pi^{\rm et}_{1}(X)$ to be surjective; this criterion is in terms of the above mentioned unique maximal locally free subsheaf associated to $\phi_*{\mathcal O}_Y$.