On a set of polarized Kähler metrics on algebraic manifolds. (English) Zbl 0706.53036

Tian, Gang

J. Differ. Geom. 32, No. 1, 99-130 (1990).

Let M be an algebraic manifold with an ample line bundle L, and let g be a Kähler metric on M which is polarized with respect to L. In this paper it is proved that the Bergmann metrics \(g_ M\) converge to g in the \(C^ 2\)-topology. This theorem is generalized to complete Kähler manifolds M with some conditions on their Ricci curvature. When M is a quasi-projective manifold the generalization is used to prove estimates involving the Ricci form and results about its extension to a smooth projective compactification of M.

Reviewer: F.Kirwan

MSC:

53C55	Global differential geometry of Hermitian and Kählerian manifolds
14C20	Divisors, linear systems, invertible sheaves

Keywords:

polarization; algebraic manifold; Kähler metric; Bergmann metrics