Mapping Problems in Serveral Complex Variables

Mapping problems deal with the property of holomorphic mappings taking a real submanifold of a complex space into another such. The interest in mapping problems originates with the work of H. Poincare in the beginning of the last century, when he observed that real hypersurfaces in 2-dimensional complex space have nontrivial holomorphic invariants. This has led to the development of a rich theory classifying Levi-nondegenerate hypersurfaces. However, the nondegeneracy of a Levi-nondegenerate hypersurface is not very natural; by this, we mean that the typical hypersurface appearing in applications of a local mapping problem result has a more singular structure. Until very recently, it was not clear how one could overcome this problem; but over the last few years, a powerful analytic approach to some of the key consequences of the classification theory has been developed (the approach is through the finite jet determination and jet parametrization problems). We propose to follow and widen this approach to the most general class of real submanifolds for which positive answers to the structure problems are possible (there is a natural kind of nondegeneracy assumption, that of holomorphic nondegeneracy, that one has to allow). Another direction we want to pursue is to study proper holomorphic maps in positive codimension, in particular, maps between balls. Very few results are known in this fascinating area, where algebra and analysis mix in the most interesting ways.

The goal of the project was the study of holomorphic mappings between real submanifolds of complex spaces. This is a geometric problem: One has a set of objects (the submanifolds) and asks what kind of structure-preserving maps between them one can find. It turns out that in our setting, often a finite set of data allows us to identify any map; in other words, our objects are quite `rigid`. In the course of the project we were able to characterize the class of manifolds (which are in addition minimal) for which such a phenomenon holds by a geometric condition (`holomorphic nondegeneracy`). While for the class of minimal manifolds this is an end to the story, we have encountered many additional questions as well as applications for the newly developed techniques.