On the Problem of Kaehler Convexity in the Bergman Metric

摘要

Let (M, ds~2) be a complete Kahler manifold of dimension n, and let H_((2))~(p,q)(M) be the space of square-integrable harmonic forms of bidegree (p,q). McNeal has studied the question: Under which reasonable conditions about the Kaehler metric can one prove the vanishing of H_((2))~(p,q)(M) when p + q ≠ n? As a sufficient condition he found that there should exist an exhausting function V for M that is at the same time a potential for ds~2 such that V dominates its gradient. We define this property as follows.