Boundary invariants and the closed range property for $\bar\partial$

This paper provides a connection between two distinct branches of research in CR geometry -- namely, analytic and geometric conditions that suffice to establish the closed range of the Cauchy-Riemann operator and CR invariants on CR manifolds. Specifically, we work on not necessarily pseudoconvex domains $\Omega\subset\mathbb{C}^n$ and define third and fourth order CR invariants on $\bd\Om$ and show that these invariants provide enough information to establish closed range for the $\bar\partial$-Laplacian in $L^2_{(0,q)}(\Omega)$ for a given, fixed $q$. The closed range estimates follow from our previously defined weak $Z(q)$ condition. We also develop powerful linear algebra machinery to translate the information from the invariants into information about the Levi form and its eigenvalues. We conclude with several examples that demonstrate the usefulness and ease of use of the new conditions.