A Lie-Theoretic Construction of Cartan-Moser Chains

Joël Merker (1)

1 LMO - Laboratoire de Mathématiques d'Orsay

Let M^3 in C^2 be a real-analytic Levi nondegenerate hypersurface. In the literature, Cartan-Moser chains are detected from rather advanced considerations: either from the construction of a Cartan connection associated with the CR equivalence problem; or from the construction of a formal or converging Poincaré-Moser normal form. This note provides an alternative direct elementary construction, based on the inspection of the Lie prolongations of 5 infinitesimal holomorphic automorphisms to the space of second order jets of CR-transversal curves. Within the 4-dimensional jet fiber, the orbits of these 5 prolonged fields happen to have a simple cubic 2-dimensional degenerate exceptional orbit, the chain locus: Sigma_0 := {(x1, y1, x2, y2) in R^4: x2 = -2*x1^2*y1 - 2*y1^3, y2 = 2*x1*y1^2 + 2*x1^3}. Using plain translations, we may capture all points by working only at one point, the origin, and computations become conceptually enlightening and simple.

Keywords: Lie prolongations of vector fields; Cauchy-Riemann manifolds; Local biholomorphic equivalences; Formal and convergent normal forms

Domains: Mathematics [math]

Joël Merker. A Lie-Theoretic Construction of Cartan-Moser Chains. Journal of Lie Theory, 2021, 31 (1), pp.029--062. ⟨hal-03286288⟩