Local boundedness of 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 $M\subset {\mathbb{C}}^n$.

References in corpus (1)

• Finite Type Points On Subsets Of ${\mathbb{C}}^n$