2021/02/28 by Biswas, Indranil, Parameswaran, A. J.
#13D07 #14F06 #14J60 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2103.00398
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 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\rm et1(Y) → π\rm et1(X) to be surjective; this criterion is in terms of the above mentioned unique maximal locally free subsheaf associated to ϕ_*\mathcal OY.