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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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[1] Bochner, S. Curvature in hermitian metric,Bull. Am. Math. Soc.,53, 179–185, (1947). · Zbl 0035.10403 · doi:10.1090/S0002-9904-1947-08778-4
[2] Borel, A.Linear Algebraic Groups, Benjamin, New York, 1969. · Zbl 0206.49801
[3] Bouche, T. Convergence de la métrique de Fubini-Study d’un fibré linéaire positif,Ann. Inst. Fourier Grenoble,40(1), 117–130,(1990). · Zbl 0685.32015 · doi:10.5802/aif.1206
[4] Bourguignon, J.-P. Géométrie riemannienne en dimension 4, (A.L. Besse), ch. VIII, Cedic-Nathan, Paris, 1981.
[5] Calabi, E. Isometric embedding of complex manifolds,Ann. Math.,58, 1–23, (1953). · Zbl 0051.13103 · doi:10.2307/1969817
[6] Serre, J.-P. Représentations linéaires et espaces homogènes kähleriens des groupes de Lie compacts, Séminaire Bourbaki 1954, Benjamin, 1966.
[7] Tian, G. On a set of polarized Kähler metrics on algebraic manifolds,J. Diff. Geom.,32, 99–130, (1990). · Zbl 0706.53036
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