Direct Proof of Termination of the Kohn Algorithm in the Real-Analytic Case

In 1979 J.J. Kohn gave an indirect argument via the Diederich-Forn\ae ss Theorem showing that finite D'Angelo type implies termination of the Kohn algorithm for a pseudoconvex domain with real-analytic boundary. We give here a direct argument for this same implication using the stratification coming from Catlin's notion of a boundary system as well as algebraic geometry on the ring of real-analytic functions. We also indicate how this argument could be used in order to compute an effective lower bound for the subelliptic gain in the $\bar\partial$-Neumann problem in terms of the D'Angelo type, the dimension of the space, and the level of forms provided that an effective \L ojasiewicz inequality can be proven in the real-analytic case and slightly more information obtained about the behavior of the sheaves of multipliers in the Kohn algorithm.Comment: 33 page

On the Relationship between D'Angelo q-type and Catlin q-type

We establish inequalities relating two measurements of the order of contact of q-dimensional complex varieties with a real hypersurface.Comment: 18 pages; accepted at the Journal of Geometric Analysis; see arXiv:1102.0356 for the origin of this investigatio

Relating Catlin and D'Angelo qq-types

We clarify the relationship between the two most standard measurements of the order of contact of q-dimensional complex varieties with a real hypersurface, the Catlin and D'Angelo $q$-types, by showing that the former equals the generic value of the normalized order of contact measured along curves whose infimum is by definition the D'Angelo $q$-type.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1302.229

Effective vanishing order of the Levi determinant

On a smooth domain in complex n space of finite D'Angelo q-type at a point, an effective upper bound for the vanishing order of the Levi determinant \text{coeff}\{\partial r \wedge \dbar r \wedge (\partial \dbar r)^{n-q}\} at that point is given in terms of the D'Angelo q-type, the dimension of the space n, and q itself. The argument uses Catlin's notion of a boundary system as well as techniques pioneered by John D'Angelo.Comment: 22 pages; typos in example from p.20 fixed in the second versio