On the Canonical Line Bundle of a Locally Hermitian Symmetric Space

Abstract

We study three problems related to towers of coverings of Hermitian symmetric spaces of non-compact type. The first one is on the possibility that the index of the canonical line bundle gets arbitrarily large on such a tower of coverings. The second one is on some vanishing and non-vanishing results on bundle-valued forms on a Kahler hyperbolic manifolds. The third one is to explain that on any tower of coverings of locally Hermitian symmetric spaces of Lie algebra type DIII, some fractional power of the canonical line bundle aK with rational \(0<a<1\) are very ample as one goes high enough in the tower.