A Theorem on Compact Complex Manifolds

Abstract

A complex manifold is the generalization from one to several variables of the abstract Riemann surface; thus an abstract Riemann surface is a complex mani- fold of complex dimension one. (For the definition of an abstract Riemann surface see (8); for the definition of a complex manifold see (1).) Such a manifold is always orientable. Connected open subsets of the space of several complex variables are examples of complex manifolds. Also algebraic varieties (without singu- larities) in a complex projective space are complex manifolds, and these are even compact. Now in the case of the compact abstract Riemann surface there exists an algebraic curve (without singularities) of which it is the Riemann surface, i.e. onto which it can be mapped analytically homeomorphically. (When we use the word analytic in this paper we always mean complex analytic.) How- ever, for complex dimension greater than one there exist compact complex mani- folds which are not analytically equivalent to any algebraic variety. Now let M be a complex manifold; by its very definition there are local com- plex coordinates in some neighborhood of each point of M, and any two admis- sible systems of local coordinates at a point are analytically related, hence a function analytic in one system is analytic in any admissible local coordinate system at that point. Thus we call a complex valued function defined on M analytic at a point x of M if it is analytic in the local coordinates at x. THEOREM I. If M is a compact complex manifold and if f is a complex function analytic at every point of M then f is a constant.