Convergent normal form and canonical connection for hypersurfaces of finite type in C2

We study the holomorphic equivalence problem for finite type hypersurfaces in C2. 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 C2.