Canonical subsheaves of torsionfree semistable sheaves
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 Y X between normal projective varieties, we give a criterion for the induced homomorphism of étale fundamental groups φ _*π ^et _1(Y) 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 _*O_Y. , number=2, journal=MATHEMATICA SCANDINAVICA, author=Biswas, Indranil and Parameswaran, A. J., year=2022, month=Jun.