Local boundedness of Catlin q-type

TL;DR

This paper generalizes Catlin's q-type for arbitrary subsets of complex space and proves the openness of the set of points with finite Catlin q-type, extending previous results on D'Angelo type.

Contribution

It introduces a new generalization of Catlin's q-type for arbitrary subsets and establishes the openness of finite q-type points for such sets.

Findings

Generalized Catlin q-type for arbitrary subsets.

Proved openness of finite Catlin q-type points.

Extended D'Angelo type results to Catlin q-type.

Abstract

In [6], D'Angelo introduced the notion of finite type for points $p$ of a real hypersurface $M$ of $\mathbb C^n$ by defining the order of contact $\Delta_q(M,p)$ of complex analytic $q$-dimensional varieties with $M$ at $p$. Later, Catlin [4] defined $q$-type, $D_q(M,p)$ for points of hypersurfaces by considering generic $(n-q+1)$-dimensional complex affine subspaces of $\mathbb C^n$. We define a generalization of the Catlin's $q$-type for an arbitrary subset $M$ of $\mathbb C^n$ in a similar way that D'Angelo's 1-type, $\Delta_1(M,p)$, is generalized in [13]. Using recent results connecting the D'Angelo and Catlin $q$-types in [1] and building on D'Angelo's work on the openness of the set of points of finite $\Delta_q$-type, we prove the openness of the set of points of finite Catlin $q$-type for an arbitrary subset $M\subset \mathbb C^n$.