The d-bar problem on Z(q) domains

Many of the most important regularity properties in the $\bar\partial$-problem (closed range, compactness, 1/2-estimates, etc) are invariant under changes of metric. In contrast, many of the known sufficient conditions which imply these properties (weak , property P, $q$-pseudoconvexity, etc) are not. In this talk, I will discuss a first attempt to bridge the gap. Specifically, I will investigate domains in a complex manifold for which the Levi form and a weight function share some common positive directions. I’ll then discuss how to build a metric in which the $L^2$ machinery applies and prove a vanishing cohomology result. These techniques also give a simpler proof that Z(q) domains satisfy 1/2 estimates and finite dimensional cohomology at the (0,q)-level. This work is joint with Debraj Chakrabarti (Central Michigan U.) and Phil Harrington (U. Arkansas).