Notions of Higher Type
Andreea C. Nicoara
TL;DR
This work compares D'Angelo's $q$-type $\Delta_q$ and Catlin's $q$-type $D_q$ for hypersurfaces in $\mathbb{C}^n$, clarifying their relationship and impact on subellipticity in the $\bar{\partial}$-Neumann problem. It presents Catlin's generic-intersection framework via the Grassmannian, elaborating the constructions $W_1$, $W_2$, and $W_3$ to isolate typical behavior and define $D_q$. A central result is that $D_q(\mathcal{I},x_0)=\tilde{\Delta}_q(\mathcal{I},x_0)$ and, for domains, $D_q(b\Omega, x_0)=\tilde{\Delta}_q(b\Omega, x_0)$, with the foundational inequality $\Delta_q\le D_q\le(\Delta_q)^{n-q+1}$. Fassina’s examples show that $\Delta_q$ and $D_q$ can differ, but a cylinder-variation technique yields equality with the tilde-versions, enabling effective subelliptic estimates. Collectively, these results unify the two notions and provide practical tools for analyzing finite $q$-type and subellipticity in smooth settings.
Abstract
Notions of finite type play an important role in several complex variables. The most standard notion is D'Angelo type, which measures the order of contact of holomorphic curves with the boundary of a domain in ${\mathbb C}^n$. For the $\bar \partial$-Neumann problem, however, the order of contact of the boundary of the domain with $q$-dimensional complex varieties controls its behavior on $(p,q)$ forms. There are two different ways of measuring this order of contact, one due to John D'Angelo and another due to David Catlin. We survey the known results about the D'Angelo and Catlin $q$-types, their relationship, and other notions that complete the picture.