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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.
32Q40 Embedding theorems for complex manifolds
32Q15 Kähler manifolds
32L20 Vanishing theorems
53C55 Global differential geometry of Hermitian and Kählerian manifolds