  A Lie algebroid is a generalization of Lie algebra that provides a general
framework to describe the symmetries of a manifold. In this paper, we
generalize the Kodaira vanishing theorem, which is a basic result in complex
geometry, to Kahler Lie algebroids. The generalization of the Kodaira vanishing
theorem states that the kernel of the Lie algebroid Laplace operator on Lie
algebroid positive line bundle-valued (p,q)-forms vanishes when p+q is
sufficiently large. The most difficult part of the proof of the generalized
Kodaira vanishing theorem is the generalization of the Kahler identities to Lie
algebroids. In this paper, we provide an approach by using local coordinate
calculation.
