Complex manifolds in q-convex boundaries

We consider a C∞ boundary bΩ⊂Cn which is q-convex in the sense that its Levi-form has positive trace on every complex q-plane. We prove that bΩ 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 1/2-subelliptic multipliers for the ∂¯-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 bΩ 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 bΩ. Altogether, the absence of complex q-manifolds in bΩ produces a subelliptic estimate on q-forms.