'On Nondegeneracy Conditions for the Levi Map in Higher Codimension'

By Francine Meylan - June 9, 2023 - session 3
Let M be a real submanifold of C N , p ∈ M and Aut(M, p) be the stability group of M at point p that is the set of (germs of) biholomorphisms F fixing p and such that F(M) ⊆ M. For a real hypersurface in complex dimension 2, H. Poincar´e initiated the study of the stability group by looking at Taylor series expansion: the condition F(M) ⊆ M means that ρ(F(z, w))|M = 0 where ρ is a defining function of M and this equation gives some constraints on the Taylor series coefficients of F. The process was carried out much later in a significant manner by Chern and Moser (1974). They proved that if M is a real-analytic hypersurface through a point p ∈ C N with non-degenerate Levi form at p and if F and G are two germs of biholomorphic maps preserving M with the same 2-jets at p then they coincide. After a historical introduction including Cartan’s Uniqueness Theorem (1931) and its consequences (the Riemann mapping Theorem fails in higher dimension!) we will present various definitions of nondegeneracy of the Levi map for real submanifolds of higher codimension in CN compare them by giving many examples and discuss the generalization to higher codimension of the above 2-jet determination theorem established by Chern and Moser.