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 Kähler manifold is weakly pseudoconvex. Thus we can regard Pic⁰(?ⁿ) as a family of weakly pseudoconvex Kähler 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 Kähler manifolds or not. Using the result of ColColţoiu, we find a 1-convex complete Kähler manifold on which the -Lemma does not hold.