Ramified covering maps and stability of pulled back bundles
Indranil Biswas, A. J. Parameswaran

TL;DR
This paper characterizes when the pullback of a stable vector bundle under a morphism between curves remains stable, establishing a precise condition involving the ramification properties of the morphism.
Contribution
It provides a necessary and sufficient condition, called genuine ramification, for the stability of pulled back bundles on algebraic curves.
Findings
Pullback of stable bundles remains stable iff the morphism is genuinely ramified.
Genuine ramification is characterized by the maximal semistable subbundle condition.
The result links ramification properties with stability preservation in vector bundles.
Abstract
Let be a nonconstant separable morphism between irreducible smooth projective curves defined over an algebraically closed field. We say that is genuinely ramified if is the maximal semistable subbundle of (equivalently, the homomorphism of etale fundamental groups is surjective). We prove that the pullback is stable for every stable vector bundle on if and only if is genuinely ramified.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology · Advanced Algebra and Geometry
