The Cauchy-Riemann equations and L^2 Serre duality on complex manifolds

In this talk I will discuss the closed range property and boundary regularity of the Cauchy-Riemann equations for domains on complex manifolds. In particular, L^2 results for the Cauchy-Riemann equations on product domains of complex manifolds and an analogue of the classical Künneth formula have been obtained recently.
I will also discuss an L^2 version of the Serre duality for domains on complex manifolds with applications to duality between harmonic and Bergman spaces.
Much of the talk is from recent joint work with Debraj Chakrabarti.