Abstract

It has been 66 years since the existence of a complex structures on Grauert Tubes was proven for the first time by Bruhat-Whitney. Still, up to now, the explicit form of such structure has remained quite mysterious to the community of experts in the field.

This is finally clarifed in a recent joint article with Bruno Salvy, where we consider complex structures with totally real zero section of the tangent bundle. We assume that the complex structure tensor is real analytic on the fibers of the tangent bundle. This hypothesis is quite natural in view of the existence result of Bruhat-Whitney.

For any torsion-free connection acting on the real analytic sections of the tangent bundle of a real analytic manifold, we provide a simple and explicit expression of the coefficients of the Taylor expansion on the fiber of the associated canonical complex structure.

An explicit global expression for the above coefficients is important for applications to analytic micro local analysis over manifolds, as it allows an explicit global construction of the complex extension of a given global Fourier integral operator defined on a real analytic manifold. The above application leads to important consequences in the theory of analytic complex deformations of a given compact complex manifold.