Complex b-manifolds

TL;DR

This paper introduces complex b-structures on manifolds with boundary, exploring their properties, associated elliptic complexes, and boundary-induced structures, with a focus on cohomology of the indicial complex of the b-Dolbeault complex.

Contribution

It defines complex b-structures on manifolds with boundary and investigates their cohomological properties and boundary-induced structures, extending the theory of complex structures to manifolds with boundary.

Findings

The complex b-structure induces an elliptic complex of b-operators.

It determines a rich boundary structure related to the complex b-structure.

The cohomology of the indicial complex of the b-Dolbeault complex is studied.

Abstract

A complex $b$-structure on a manifold $\M$ with boundary is an involutive subbundle $\bT^{0,1}\M$ of the complexification of $\bT\M$ with the property that $\C\bT\M = \bT^{0,1}\M + \bar{\bT^{0,1}\M}$ as a direct sum; the interior of $\M$ is a complex manifold. The complex $b$-structure determines an elliptic complex of $b$-operators and induces a rich structure on the boundary of $\M$. We study the cohomology of the indicial complex of the $b$-Dolbeault complex.