Genuinely ramified maps and pseudo-stable vector bundles

2023/02/15 by Biswas, Indranil, Parameswaran, A. J.
#13D07 #14E20 #14F06 #14J60 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2302.07463

Abstract

Let X and Y be irreducible normal projective varieties, of same dimension, defined over an algebraically closed field, and let f : Y → X be a finite generically smooth morphism such that the corresponding homomorphism between the étale fundamental groups f_*:π^{\rm et}₁(Y) →π^{\rm et}₁(X) is surjective. Fix a polarization on X and equip Y with the pulled back polarization. For a point y₀∈ Y, let \varpi(Y, y₀) (respectively, \varpi(X, f(y₀))) be the affine group scheme given by the neutral Tannakian category defined by the strongly pseudo-stable vector bundles of degree zero on Y (respectively, X). We prove that the homomorphism \varpi(Y, y₀) → \varpi(X, f(y₀)) induced by f is surjective. Let E be a pseudo-stable vector bundle on X and F ⊂ f^*E a pseudo-stable subbundle with μ(F)= μ(f^*E). We prove that f^*E is pseudo-stable and there is a pseudo-stable subbundle W ⊂ E such that f^*W = F as subbundles of f^*E.

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