Lie groupoid Riemann-Roch-Hirzebruch theorem and applications

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 introduce Lie algebroid index theory and study the Lie algebroid Dolbeault operator. We also introduce Connes' index theory on regular foliated manifolds to obtain a generalized Riemann-Roch theorem on manifolds with regular foliation. We show that the topological side of Connes' index theory can be identified with the topological side of Lie algebroid index theory. Finally, we introduce Lie algebroid Kodaira vanishing theorem, and provide some applications and examples. The Lie algebroid Kodaira vanishing theorem can be used on the analytic side of Connes' theorem to attest to the criterion of a positive line bundle from its topological information.