# The Bergman Kernel on Forms: General Theory

## TL;DR

This paper investigates the properties and construction of the Bergman kernel on differential forms, revealing fundamental differences from the function case and providing explicit computations for specific forms and domains.

## Contribution

It demonstrates the failure of basic properties for the Bergman kernel on forms and offers a detailed construction and explicit formulas, especially for (0,n-1)-forms.

## Key findings

- Pointwise evaluation fails for (0,q)-forms, q ≥ 1.
- Explicit Bergman kernel for (0,n-1)-forms is derived.
- Bergman kernel size on (0,1)-forms in the ball is not controlled by the metric.

## Abstract

The goal of this note 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 $(0,q)$-forms, $q \geq 1$. 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 Bergman kernel on functions.