UC San Diego

We investigate the geometric property of transversality of holomorphic, formal or CR mappings between real-analytic, formal or smooth generic submanifolds of complex spaces of equidimension as well as of different dimensions. In Chapter 3, we shall consider the CR transversality in equidimension case. The main purpose of this chapter is to show that a holomorphic, formal or smooth CR mapping sending a real-analytic, smooth or formal generic submanifold M into such another Mʹ is CR transversal to the target, provided that the source manifold is of finite bracket type and the mapping is of generic full rank. This result and its corollary completely resolve two questions posed by Peter Ebenfelt and Linda Preiss Rothschild in a paper from 2006. We also show that under a very mild assumption on the source manifold, the generic full rank condition imposed on the mapping is also necessary for the CR transversality to hold. This result confirms a conjecture in a paper by Bernhard Lamel and Nordine Mir. In Chapter 4, we consider the transversality of mappings when the target manifold is of higher dimension. We will restrict ourself to the situation in which both manifolds M and Mʹ are hypersurfaces in Cn⁺¹ and CN⁺¹ respectively, where 1 < n < N. A main result of this chapter implies that under certain restrictions on the dimensions n, N and the rank of the Levi-form of the target hypersurface, if the set of points at which the mapping H fails to be a local embedding has codimension at least 2, then the mapping must be transversal to the target at all points. Another result of this chapter implies that under some more restrictive assumptions, any finite holomorphic mapping sending M into Mʹ is transversal at all points, unless the source hypersurface is of infinite type. This result may be considered as a different dimension analogue of a theorem by M. Salah Baouendi and Linda Preiss Rothschild from 1990