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Embedding of a compact Kähler manifold into complex projective space. (English) Zbl 1035.32017
From the introduction: We consider a restricted class of Kähler manifolds defined by a certain topological condition. K. Kodaira’s well known embedding theorem [see e.g. Ann. Math. (2) 60, 28–48 (1954; Zbl 0057.14102)] can be formulated as follows: $$A$$ compact complex analytic manifold $$M$$ is projective algebraic if and only if there exists a positive line bundle on it.
In this paper we show that the assumption on the line bundle may be weakened if we replace it by the canonical line bundle $$K$$ of $$M$$. According to [S.-T. Yau, Commun. Pure Appl. Math. 31, 339–411 (1978; Zbl 0369.53059), Theorem 2], on a compact Kähler manifold $$M$$ any condition on the positivity (resp. negativity) of the first Chern class of $$M$$ is equivalent to the same condition on the positivity (resp. negativity) of the Ricci curvature of $$M$$. In view of this results and the fact that the first Chern class of $$M$$ is equal to the Chern class of the dual line bundle $$K^*$$ of $$K$$, we prove the following embedding theorem: Theorem. Let $$M$$ be a compact connected Kähler manifold of complex dimension $$n$$, and $$\mu_i$$ $$(i=1,2, \dots,n)$$ be the eigenvalues of the Ricci curvature of $$M$$ with respect to the Kähler metric $$ds^2$$ on $$M$$. Suppose that $$\mu_i+\mu_j\geq 0$$ at each point of $$M$$ and $$\mu_i+ \mu_j\leq 0$$ at one point of $$M$$ for any $$i,j$$, such that $$1\leq i<j\leq n$$, then $$M$$ is projective algebraic.
##### MSC:
 32Q40 Embedding theorems for complex manifolds 32Q15 Kähler manifolds 32L20 Vanishing theorems 53C55 Global differential geometry of Hermitian and Kählerian manifolds
##### Keywords:
Kähler manifolds; embedding theorem; projective algebraic