On the -equation¶over pseudoconvex Kähler manifolds

Abstract

The Picard variety Pic0(? n ) of a complex n-dimensional torus? n is the group of all holomorphic equivalence classes of topologically trivial holomorphic (principal) line bundles on ? n . The total space of a topologically trivial holomorphic (principal) line bundle on a compact Kahler manifold is weakly pseudoconvex. Thus we can regard Pic0(? n ) as a family of weakly pseudoconvex Kahler manifolds. We consider a problem whether the Kodaira's -Lemma holds on a total space of holomorphic line bundle belonging to Pic0(? n ). We get a criterion for this problem using a dynamical system of translations on Pic0(? n ). We also discuss the problem whether the -Lemma holds on strongly pseudoconvex Kahler manifolds or not. Using the result of ColColţoiu, we find a 1-convex complete Kahler manifold on which the -Lemma does not hold.