An embedding theorem of complete Kähler manifolds of positive Ricci curvature onto quasi-projective varieties
It is well known that a compact Kähler manifold with positive Ricci curvature is projective algebraic. As a non compact version, the author proves the following Theorem. Let X be an n-dimensional complete Kähler manifold of positive Ricci curvature. Suppose for some base point \(x_ 0\) and some positive constants C and c we have (i) scalar curvature \(\leq C/d(x_ 0;x)^ 2\), \(\int_{x}Ric^ n<\infty\); (ii) \(Vol(B(x_ o;r))\geq cr^{2n}\), where \(d(x_ 0,x)\) denotes the geodesic distance and \(B(x_ 0;r)\) denotes the geodesic ball centered at \(x_ 0\) of radius r. Then, X is biholomorphic to a quasi-projective variety. Previously the author has proved the above theorem under the assumption that the holomorphic bisectional curvature is positive. The author follows a similar scheme of proof for the case of positive biholomorphic sectional curvature. However, he has needed much more arguments due to the lack of a single plurisubharmonic function for the general case.
scientific article; zbMATH DE number 4169620
scientific article