Cohomology and splitting of Hermitian-Einstein vector bundles (Q805820)

Using a cohomology vanishing theorem the following result is proved. Let E be a Hermitian-Einstein vector bundle of rank r on \({\mathbb{C}}{\mathbb{P}}^ n\) and let \(I=\{R(e,\bar e,t,\bar t):\) \(e\in E_ x\), \(t\in T_ x\), \(| e| =| t| =1\), \(x\in {\mathbb{C}}{\mathbb{P}}^ n\}\), where R is the curvature tensor. If \(n\geq r^ 2+1\) and \[ \sup I-\inf I<(n-r^ 2)/r^ 2(r^ 2+1), \] then E splits, i.e. E(\(\mu\)) is trivial for some integer \(\mu\).