Bounds on the dimension of \(L^ 2\) holomorphic sections of vector bundles over complete Kähler manifolds of finite volume (Q1071143)

Let X be a complete Kähler manifold of complex dimension n and finite volume. In this article the author obtains estimates for the dimension of the space of \(L^ 2\)-holomorphic sections of the pth tensor power of a hermitian vector bundle E of rank k over X. If the curvature of E is bounded from above then the dimension is at most \(Ck^ pp^ n\) where C is a positive constant depending only on X and E. If the curvature of E is bounded from above and below by positive constants and the Ricci curvature of X is bounded from below then for p large enough there is a lower bound on the dimension of the same form \(Ck^ pp^ n\). The author hopes to use these estimates to study compactifications of complete Kähler manifolds of finite volume.