On vector bundles over hyperkähler twistor spaces

We study the holomorphic vector bundles E over twistor space $${{\,\mathrm{Tw}\,}}(M)$$ of a compact simply connected hyperkähler manifold M. give characterization semistability condition for in terms its restrictions to sections projection $$\pi \,:\, {{\,\mathrm{Tw}\,}}(M)\,\longrightarrow \, {\mathbb {CP}}^1$$ . It is shown that if admits connection, then holomorphically trivial and connection on as well. For any irreducible bundle prime rank, we prove restriction generic fibre $$ stable. On other hand, K3 surface M, construct examples composite rank whose every non-stable. have obtained new method constructing spaces; this employed these examples.