Alekseevskij, D. V.; Perelomov, A. M. Kähler-Einstein metrics in holomorphic bundles. (English. Russian original) Zbl 0635.53050 Funct. Anal. Appl. 21, No. 1-3, 144-146 (1987); translation from Funkts. Anal. Prilozh. 21, No. 2, 66-67 (1987). 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 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 \((Ric(\tilde g)=\mu \tilde g)\). Reviewer: B.Apanasov MSC: 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32M99 Complex spaces with a group of automorphisms 32L05 Holomorphic bundles and generalizations Keywords:Hermitian vector bundle; Kähler-Einstein metric Citations:Zbl 0431.53056 PDFBibTeX XMLCite \textit{D. V. Alekseevskij} and \textit{A. M. Perelomov}, Funct. Anal. Appl. 21, No. 1--3, 144--146 (1987; Zbl 0635.53050); translation from Funkts. Anal. Prilozh. 21, No. 2, 66--67 (1987) Full Text: DOI References: [1] E. Calabi, Ann. Sci. École Norm. Sup.,12, 269-294 (1979). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.