$$L^2$$ L 2 estimates for the $$\bar{\partial }$$ ∂ ¯ operator

Abstract

We present the theory of twisted L 2 estimates for the Cauchy-Riemann operator and give a number of recent applications of these estimates. Among the applications: extension theorem of Ohsawa-Takegoshi type, size estimates on the Bergman kernel, quantitative information on the classical invariant metrics of Kobayshi, Caratheodory, and Bergman, and sub elliptic estimates on the∂-Neumann problem. We endeavor to explain the flexibility inherent to the twisted method, through examples and new computations, in order to suggest further applications.