Forms and Chern Classes on Hermitian Lie Algebroids

In this paper, we study Hermitian Lie algebroids and introduce the notion of Chern A -connection on Hermitian vector bundles. Then, we prove that on every holomorphic vector bundle there exists a unique Chern A -connection and find its connection form and curvature form. Also, we generalize Weitzenböck’s formula to complex Lie algebroids and prove vanishing theorem for holomorphic sections. Moreover, we extend the Chern classes in complex geometry to Lie algebroids framework.