The Taylor Expansion of CR Germs on a Class of Levi Degenerate Models

Given the ring \(\mathrm {CR}^\infty (M,0)\) of germs of smooth CR functions defined on a real submanifold M of \(\mathbb C^n\) , the Borel map \(\mathfrak b_0\) is the map assigning to a germ \(f\in \mathrm {CR}^\infty (M,0)\) its formal Taylor expansion \(\mathfrak b_0(f)\) at 0. The image of the Borel map is a subalgebra of the ring of formal power series \(\mathbb C[[z_1,\ldots ,z_n]]\) : However, the general structure of the image is not yet well understood. In all examples studied so far, the image is given by the tensor product of a ring of formal series with a ring of convergent series. This chapter constructs examples for which the structure of the image is more complicated: As previously conjectured, this is linked to the presence of singular complex varieties inside of M.