Complex manifolds in $q$-convex boundaries

Abstract

We consider a $C^\infty$ boundary $b\Omega \subset {\mathbb {C}}^n$ which is $q$-convex in the sense that its Levi-form has positive trace on every complex $q$-plane. We prove that $b\Omega$ is tangent of infinite order to the complexification of each of its submanifolds which is complex tangential and of finite bracket type. This generalizes Diederich and Fornaess (Ann Math 107:371–384, 1978) from pseudoconvex to $q$-convex domains. We also readily prove that the rows of the Levi-form are $\frac{1}{2}$-subelliptic multipliers for the $\bar{\partial }$-Neumann problem on $q$-forms (cf. Ho in Math Ann 290:3–18, 1991). This allows to run the Kohn algorithm of Acta Math 142:79–122 (1979) in the chain of ideals of subelliptic multipliers for $q$-forms. If $b\Omega$ is real analytic and the algorithm gets stuck on $q$-forms, then it produces a variety of holomorphic dimension $q$, and in fact, by our result above, a complex $q$-manifold which is not only tangent but indeed contained in $b\Omega$. Altogether, the absence of complex $q$-manifolds in $b\Omega$ produces a subelliptic estimate on $q$-forms.