On Automorphisms of Complex \(b^k\) -Manifolds

The b-calculus of Melrose is a tool for studying structures on a smooth manifold with a first-order degeneracy at a given hypersurface. In this framework, Mendoza defined complex b-manifolds. In the spirit of work of Scott, we extend Mendoza’s definition to the case of higher-order degeneracies, introducing the notion of a complex \(b^k\) -manifold for k a positive integer. We then investigate the local and global automorphisms of complex \(b^k\) -manifolds. We also propose \(b^k\) -analogues for some classical spaces of holomorphic functions.