Families of stable bundles on the fibres of the hyperkähler twistor projection

2019/08/14 by Tomberg, Artour
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1908.05333

Abstract

Given a holomorphic vector bundle E on the twistor space Tw(M) of a simple hyperkähler manifold M, we view it as a family of bundles \E_I\ on the fibres π^-1(I) of the twistor projection π: Tw(M) → \mathbbCP¹, and study the relationship between stability of E and its fibrewise stability. We verify that the argument of Teleman establishing the Zariski openness of stability and semi-stability in families of bundles applies in the case of the family \E_I\. We prove a partial converse to a result of Kaledin and Verbitsky, showing that an irreducible bundle E on Tw(M) is generically fibrewise stable if the rank of E is 2 or 3, or at least one element of the family \E_I\ is a simple bundle, in the sense that Hom(E_I, E_I) = ℂ.

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