∂¯-tangential invariants of certain vector bundles over complex foliations

The Dolbeault cohomology plays an important role in the study of some ∂¯-invariants of complex and holomorphic vector bundles as ∂¯-Chern classes and Atiyah classes. In this paper we generalize similar invariants and their properties in the tangential Dolbeault cohomology. More exactly, we introduce and we study tangential Atiyah classes for F-holomorphic vector bundles and ∂¯-tangential Chern classes for tangentially smooth vector bundles over a manifold M endowed with a complex foliation F. Also, ∂¯-tangential secondary invariants are studied following similar constructions for Lie algebroids. The notions are introduced by a global formalism that is used in the tangential theory of foliated spaces.