### References & Citations

# Mathematics > Algebraic Geometry

# Title: Canonical subsheaves of torsionfree semistable sheaves

(Submitted on 28 Feb 2021 (v1), last revised 7 Jun 2021 (this version, v2))

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 $\phi_*{\mathcal O}_Y$.

## Submission history

From: Indranil Biswas [view email]**[v1]**Sun, 28 Feb 2021 05:24:49 GMT (10kb)

**[v2]**Mon, 7 Jun 2021 09:14:42 GMT (11kb)

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