Direct Proof of Termination of the Kohn Algorithm in the Real-Analytic Case

TL;DR

This paper provides a direct proof that finite D'Angelo type guarantees the termination of the Kohn algorithm for real-analytic pseudoconvex domains, using stratification and algebraic geometry techniques.

Contribution

It introduces a direct argument leveraging boundary stratification and algebraic geometry to establish termination of the Kohn algorithm in the real-analytic case, improving upon previous indirect methods.

Findings

Direct proof of Kohn algorithm termination for real-analytic boundaries.

Potential to compute effective lower bounds for subelliptic gain.

Use of algebraic geometry and boundary stratification in complex analysis.

Abstract

In 1979 J.J. Kohn gave an indirect argument via the Diederich-Forn\ae ss Theorem showing that finite D'Angelo type implies termination of the Kohn algorithm for a pseudoconvex domain with real-analytic boundary. We give here a direct argument for this same implication using the stratification coming from Catlin's notion of a boundary system as well as algebraic geometry on the ring of real-analytic functions. We also indicate how this argument could be used in order to compute an effective lower bound for the subelliptic gain in the $\bar\partial$-Neumann problem in terms of the D'Angelo type, the dimension of the space, and the level of forms provided that an effective \L ojasiewicz inequality can be proven in the real-analytic case and slightly more information obtained about the behavior of the sheaves of multipliers in the Kohn algorithm.