Kodaira vanishing theorems for Kahler Lie algebroids

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.