Approximation Theorems in Algebra and CR Geometry

In this project, we study whether a formal holomorphic map between real objects in a complex space can be approximated by convergent holomorphic maps. This question is related to Michael Artin`s Theorem that every formal solution to a system of holomorphic equations can be approximated by convergent solutions. However, there are obstructions to the approximation property in the setting which we propose to study: There exist real submanifolds of complex spaces which are formally equivalent but no holomorphic map takes one of them into the other. The obstructions are quite subtle and involve geometric properties of the real submanifolds in question. Typically, one has to deal with three structures (the CR structure, the type structure, and the nondegeneracy structure) which might all degenerate in various ways. Positive results show that in simple cases (i.e. when all or some of the structures are in a suitable sense uniform), the approximation property holds. However, in more sophisticated settings, it is not known whether approximation is possible or not. The study of this question leads to interesting questions in algebraic geometry, which we will take up in this project, which will involve researchers from algebraic geometry and CR geometry.

A famous theorem of Artin says that every formal solution of a system of analytic equations can be approximated by analytic solutions of that system. In the theory of Cauchy-Riemann manifolds, which originated with Poincaré, there are surprising applications of Artins theorem, which imply similar results for solutions of certain partial differential equations. The project aimed on the one hand at making the theory of approximation accessible to researchers in Cauchy-Riemann geometry and on the other hand, at understanding the geometric results in an algebraic context.In the course of the project, we developed a thorough geometric understanding of the theory of the approximation theorem; in particular, points at which the solution set of a system of analytic equations has a nice form were characterized in a geometric manner. Some of the questions we encountered were brought to a negative resolution: For example, we were able to show that there are (more general) systems of equations which arise in the applications, which have the property that even though they do not possess any analytic solutions, they permit a formal solution.