On a conjecture of J. M. Lee (Q1326591)

The following conjecture of J. M. Lee is discussed: compact strictly pseudo-convex \(CR\) manifolds whose \(CR\) structure has a vanishing first Chern class admit a global pseudo-Einstein structure. Using the parabolic dilations of the Heisenberg group the author constructs a family \(H_ n(s)\), \(0 < s < 1\) of compact strictly pseudo-convex \(CR\) manifolds, so that each \(H_ n(s)\) satisfies the Lee conjecture. Endowing \(H_ n(s)\) with a contact form \(\theta\) he proves: Theorem 2. Let \(0 < s < 1\) and \(n > 1\). Then i) all Chern classes of \({\mathcal H}_ s = T_{1,0}(H_ n(s))\) vanish, and ii) the contact form \(\theta\) is pseudo-Einstein and has non-vanishing pseudo-Hermitian torsion. Also the Lee conjecture for compact strictly pseudo-convex \(CR\) manifolds with a regular contact vector is solved affirmatively.