Weighted L2 Holomorphic Functions on Ball-Fiber Bundles Over Compact Kahler Manifolds

초록

Let (M) over tilde be a complex manifold, Gamma be a torsion-free cocompact lattice of Aut((M) over tilde) and rho : Gamma -> SU(N, 1) be a representation. Suppose that there exists a rho-equivariant totally geodesic isometric holomorphic embedding tau : (M) over tilde -> B-N. Let M := (M) over tilde/Gamma and Sigma := B-N /rho(Gamma). In this paper, we investigate a relation between weighted L-2 holomorphic functions on the fiber bundle Omega := M x (rho) B-N and the holomorphic sections of the pull-back bundle tau* ((ST Sigma)-T-m*) over M. In particular, A(alpha)(2)(Omega) has infinite dimension for any alpha > -1 and if n < N, then A(-1)(2)(Omega) also has the same property. As an application, if Gamma is a torsion-free cocompact lattice in SU(n, 1), n >= 2, and rho : Gamma -> SU(N, 1) is a maximal representation, then for any alpha > -1, A(alpha)(2)(B-n x B-rho(N)) has infinite dimension. If n < N, then A(-12)( B-n x (rho) B-N) also has the same property.

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