Convergent normal form and canonical connection for hypersurfaces of finite type in ${\mathbb{C}}^2$

Abstract

We study the holomorphic equivalence problem for finite type hypersurfaces in ${\mathbb{C}}^2$. We discover a geometric condition, which is sufficient for the existence of a natural convergent normal form for a finite type hypersurface. We also provide an explicit construction of such a normal form. As an application, we construct a canonical connection for a large class of finite type hypersurfaces. To the best of our knowledge, this gives the first construction of an invariant connection for Levi-degenerate hypersurfaces in ${\mathbb{C}}^2$.

The paper in accepted to Advances in Mathematics. In this version, we revised the construction by normalizing the complex defining function of a real hypersurface instead of the real defining function. This is due to a remark of Martin Kolar who observed that using the real defining function leads to a singular ODE for a normalizing transformation in general