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Kähler-Einstein metrics and projective embeddings. (English) Zbl 1057.53032
One can naturally ask when a projective manifold can also be realized as a complex isometric submanifold with respect to the Fubini-Study metric. E. Calabi [Ann. Math. (2) 58, 1–23 (1953; Zbl 0051.13103)] showed that this is not possible if the manifold has constant non-positive holomorphic sectional curvature. The present author shows that, more generally, any compact Einstein complex submanifold $$(M,g)\hookrightarrow (\mathbb{P}^N, g_{F-S})$$ has ctrictly positive scalar curvature. The proof, by absurd, uses the analyticity of the Einstein submanifold and consists in finding, locally, a special coordinate system and a preferred potential which allows to estimate $$\text{vol}(M,g)$$ as bigger than the volume of a certain subset which is shown to be Zariski dense in $$\mathbb{C}^n$$.

##### MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32Q20 Kähler-Einstein manifolds 32Q25 Calabi-Yau theory (complex-analytic aspects)
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