38. On Kahler Varieties of Restricted Type (An Intrinsic Characterization of Algebraic Varieties)

Abstract

A compact complex analytic variety V is called a Kdhler variety of restricted type' or a Hodge variety2 if V carries a Kahler metric ds2 = 2 Zga,(dzadz) such that the associated exterior form co = i2ZgadZadz belongs to the cohomology class of an integral 2-cocycle on V. In what follows such a metric will be called a Hodge metric. The main purpose of the present paper is to prove that every Hodge variety is (bi-regularly equivalent to) a non-singular algebraic variety imbedded in a projective space.3 Since every non-singular algebraic variety in a projective space carries a Hodge metric, our main result may also be stated in the following form: A compact complex analytic variety V is (bi-regularly equivalent to) a nonsingular algebraic variety imbedded in a projective space if and only if V can carry a Hodge metric. This paper4 is divided into four sections. In Section 1 we give a summary of some known results concerning complex line bundles over Kdhler varieties. Section 2 is concerned with quadratic transformations. The proof of our main theorem (Theorem 4) is given in the following Section 3. The final Section 4 is devoted to several applications of our main theorem. In particular, we prove that a compact complex analytic variety is a non-singular algebraic variety imbedded in a projective space if it carries a Hermitian metric whose Ricci curvature is everywhere negative [or positive] definite (Theorem 5). We also derive the following result: Let 6B be a bounded domain in the space of n complex variables and let A be a discontinuous group of analytic automorphisms without fixed points of 6B such that 6(/A is compact. Then the factor space 6e/A is a non-singular algebraic variety imbedded in a projective space (Theorem 6). The author wishes to express his sincere thanks to A. Borel for valuable suggestions given during the period of the preparation of this paper.