The Bergman kernel on forms: General theory

The goal of this paper is to explore the Bergman projection on forms. In particular, we show that some of most basic facts used to construct the Bergman kernel on functions, such as pointwise evaluation in $L^2_{0,q}(\Omega )\cap \ker \bar \partial _q$, fail for $(p,q)$-forms, $q \geq 1$, $p\geq 0$. We do, however, provide a careful construction of the Bergman kernel and explicitly compute the Bergman kernel on $(0,n-1)$-forms. For the ball in $\mathbb{C}^2$, we also show that the size of the Bergman kernel on $(0,1)$-forms is not governed by the control metric, in stark contrast to the Bergman kernel on functions.