Criteria for quasi-projectivity

Abstract

In transcendental algebraic geometry it often happens that one gets complex spaces by analytic operations and one wishes to show that they are actually algebraic. In the case of compact complex manifolds and spaces, one has Kodaira's embedding theorem and Grauert's generalization respectively. Proposition I, which is a natural generalization of Kodaira's theorem, gives differential geometric criteria for the quasi-projectivity of certain non-compact manifolds. One corollary of it that gives its flavour is: Corollary. Let X be Zariski open in a compact complex manifold 2g. Let L be a holomorphic line bundle on X that has a Hermitian structure over X with only L 2 poles at infinity and whose curvature form is the Hermitian form associated to a complete metric on X. Then X is Moisezon and X is quasi-projective with respect to the algebraic structure induced J?om f2. The definition of L 2 poles at infinity is given after Proposition I. It is noteworthy that no upper bounds on the growth of the curvature of the Hermitian structure are required! Section I is devoted to Proposition I and some immediate corollaries. The form of Proposition I was dictated by its application to the study of the quasi-projectivity of the image of the Griffiths-Riemann period mapping that arises in algebraic geometry. One can look at [4-7, 9, 10, 19] for background on the period mapping; the above problem is discussed in [7, pp. 259ff.] and [6 (III), Appendix C]. In Section II it is shown that the period mapping satisfies the conditions of Proposition I when the mapping is proper and has a manifold as an image.