2020/01/30 by Joel Merker, Merker, Joel
Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #math.CV #math.DG
paper · pdf · doi:10.48550/arxiv.2001.11276
This work was supported in part by the Polish National Science Centre (NCN) via the grant number 2018/29/B/ST1/02583
arxiv created 2020/07/08 · arxiv updated 2020/07/09
Let M3 ⊂ ℂ2 be a Cω 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: Σ0 := \ (x1,y1,x2,y2) ∈ ℝ4 \colon x2 = -2x12y1-2y13, y2 = 2x1y12 + 2x13 \. By plain translations, we may capture all points by working only at one point, the origin, and computations, although conceptually enlightening, become disappointingly simple.