Thesis Defense: A generalized Kodaira vanishing theorem for Lie algebroids and a Riemann–Roch theorem for singular foliation

Abstract: A Lie algebroid is a generalization of Lie algebra that provides a general framework to describe the symmetries of a manifold. In this thesis, we first generalize the Kodaira vanishing theorem, which is a basic result in complex geometry, to K\"{a}hler 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 K\"{a}hler identities to Lie algebroids. Secondly, 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. Thirdly, we 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. For another application, we study K\"{a}hler b-manifolds and introduce a generalized Kodaira vanishing theorem on K\"{a}hler b-manifolds.

Host: Xiang Tang