A twisted $\bar{\partial}_f$-Neumann problem and Toeplitz $n$-tuples from singularity theory

TL;DR

This paper introduces a twisted $ar{ ext{}}{ ext{d}}_f$-Neumann problem linked to singularity theory, connecting it to Toeplitz $n$-tuples and explicitly computing related cohomology on Bergman spaces.

Contribution

It establishes a new twisted $ar{ ext{}}{ ext{d}}_f$-Neumann problem and links it to the Koszul complex for Toeplitz $n$-tuples, providing explicit cohomology calculations.

Findings

Solved the $ar{ ext{}}{ ext{d}}_f$-Neumann problem for singularities.

Connected the problem to the Koszul complex for Toeplitz $n$-tuples.

Explicitly computed the cohomology of the $L^2$ holomorphic Koszul complex.

Abstract

A twisted $\bar{\partial}_f$-Neumann problem associated to a singularity $(\mathscr{O}_n,f)$ is established. By constructing the connection to the Koszul complex for toeplitz $n$-tuples $(f_1,\cdots,f_n)$ on Bergman spaces $B^0(D)$, we can solve this $\bar{\partial}_f$-Neumann problem. Moreover, we can compute the cohomology of the $L^2$ holomorphic Koszul complex $(B^*(D),\partial f\wedge)$ explicitly