Abstract
Let [Formula: see text] be a separable finite surjective map between irreducible normal projective varieties defined over an algebraically closed field, such that the corresponding homomorphism between étale fundamental groups [Formula: see text] is surjective. Fix a polarization on [Formula: see text] and equip [Formula: see text] with the pullback, by [Formula: see text], of this polarization on [Formula: see text]. Given a stable vector bundle [Formula: see text] on [Formula: see text], we prove that there is a vector bundle [Formula: see text] on [Formula: see text] with [Formula: see text] isomorphic to [Formula: see text] if and only if the direct image [Formula: see text] contains a stable vector bundle [Formula: see text] such that [Formula: see text] We also prove that [Formula: see text] is stable for every stable vector bundle [Formula: see text] on [Formula: see text].
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