Global ${\bar{\partial }}_{\mathbf{M}}$-homotopy with $C^k$ estimates for a family of compact, regular q-pseudoconcave CR manifolds

Abstract

Let ${\mathbf{M}}_0$ be a compact, regular q-pseudoconcave compact CR submanifold of a complex manifold $\mathbf{G}$ and $\mathcal{B}$ - a holomorphic vector bundle on $\mathbf{G}$ such that $\dim H^r({\mathbf{M}}_0,\mathcal{B}{|}_{\mathbf{M}})=0$ for some fixed $r<q$. We prove a global homotopy formula with $C^k$ estimates for $r$-cohomology of $\mathcal{B}$ on arbitrary CR submanifold $\mathbf{M}$ close enough to ${\mathbf{M}}_0$.

39 pages