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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
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##### References:
 [1] E. Calabi, Ann. Sci. École Norm. Sup.,12, 269-294 (1979).
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