Kähler-Einstein metrics in holomorphic bundles (Q1097532)

Let (E,h) be a Hermitian vector bundle associated with a Hodge metric g on a complex manifold M; i.e., \(E=L^{m_ 1}(g)\oplus...\oplus L^{m_ r}(g)\) and the \(L^{m_ i}(g)\) are holomorphic vector bundles of degree \(m_ i\). Following \textit{E. Calabi} [Ann. Sci. Ec. Norm. Super., IV. Ser. 12, 269-294 (1978; Zbl 0431.53056)], the authors define for h some Kähler metric \(\tilde g\) on E and obtain for \(m_ 1=...=m_ r=m\) necessary and sufficient conditions when this becomes a Kähler-Einstein metric \((Ric(\tilde g)=\mu \tilde g)\).