Regular types and order of vanishing along a set of non-integrable vector fields

TL;DR

This paper surveys recent progress on Bloom's conjecture related to regular types and D'Angelo type in complex analysis, and proves a fundamental property on the vanishing order of functions along non-integrable vector fields.

Contribution

It provides a comprehensive survey of Bloom's conjecture and introduces a new proof of a key property of vanishing order using advanced normalization theorems.

Findings

Progress on Bloom's conjecture in complex dimension ≥ 4

A new proof of vanishing order property for non-integrable vector fields

Clarification of the relationship between regular types and D'Angelo type

Abstract

This paper has two parts. We first survey recent efforts on the Bloom conjecture which still remains open in the case of complex dimension at least 4. Bloom's conjecture concerns the equivalence of three regular types. There is a more general important notion, called the singular D'Angelo type (or simply, D'Angelo type). While the finite D'Angelo type condition is the right one for the study of local subelliptic estimates for Kohn's $\overline{\partial}$-Neumann problem, regular types are important as their finiteness gives the global regularity up to the boundary of solutions of Kohn's $\overline{\partial}$-Neumann problem. In the second part of the paper, we provide a proof of a seemingly elementary but a truly fundamental property (Theorem 2.2 or its CR version Theorem 2.5) on the vanishing order of smooth functions along a system of non-integrable vector fields. A special case, Corollary 2.6, of Theorem 2.5 had already appeared in a paper of D'Angelo. A main goal in this part is to provide proofs for these results for the purpose of future references. Our arguments are based on a deep normalization theorem for a system of non-integrable vector fields due to Helffer-Nourrigat, as well as its late generalization in Boauendi-Rothschild.