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

Abstract

Given a holomorphic vector bundle $E$ on the twistor space $\mathrm{Tw}(M)$ of a simple hyperkähler manifold $M$, we view it as a family of bundles $\left\{E_I\right\}$ on the fibres ${\pi }^{-1}(I)$ of the twistor projection $\pi :\mathrm{Tw}(M)\to {\mathbb{CP}}^1$, 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 $\left\{E_I\right\}$. We prove a partial converse to a result of Kaledin and Verbitsky, showing that an irreducible bundle $E$ on $\mathrm{Tw}(M)$ is generically fibrewise stable if the rank of $E$ is 2 or 3, or at least one element of the family $\left\{E_I\right\}$ is a simple bundle, in the sense that $\mathrm{Hom}(E_I,E_I)=\mathbb{C}$.

29 pages