On vector bundles over hyperkähler twistor spaces

2019/11/03 by Biswas, Indranil, Tomberg, Artour
#32L10 #32L25 #53C28 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1911.00833

Abstract

We study the holomorphic vector bundles E over the twistor space Tw(M) of a compact simply connected hyperkähler manifold M. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection π:Tw(M)→ CP¹. It is shown that if E admits a holomorphic connection, then E is holomorphically trivial and the holomorphic connection on E is trivial as well. For any irreducible vector bundle E on Tw(M) of prime rank, we prove that its restriction to the generic fibre of πis stable. On the other hand, for a K3 surface M, we construct examples of irreducible vector bundles of any composite rank on Tw(M) whose restriction to every fibre of πis non-stable. We have obtained a new method of constructing irreducible vector bundles on hyperkähler twistor spaces; this method is employed in constructing these examples.

Related

• Families of stable bundles on the fibres of the hyperkähler twistor projection
• An example of a stable but fibrewise nonstable bundle on the twistor\n space of a hyperk "ahler manifold
• Hypercomplex structures arising from twistor spaces
• Stable bundles on Hopf manifolds
• The universal connection for principal bundles over homogeneous spaces and twistor space of coadjoint orbits