The canonical Kähler potential on the parameter space of the versal family of CR structures (Q1765784)

Let \((M,^{0}T'')\) be a compact strongly pseudo convex CR structure with \(\dim_\mathbb{R}=2n-1\). Then, in [Mich. Math. J. 50, No.~3, 517--549 (2002; Zbl 1065.32018)], by \textit{T. Akahori, P. M. Garfield} and \textit{J. M. Lee}, the construction of the versal family of CR structures is settled. The purpose of this paper is to introduce a canonical Kähler metric for the parameter space of this versal family if the CR structure admits a normal vector field and a non-vanishing CR \(n\)-form with the condition \(H^1(M, \mathcal{O}_M)=0\) and \[ H^1(M, \wedge^{n-1}(T')^*)\simeq Z^1 \, , \] where \(Z^1\) is introduced in [Compos. Math. 85, 57--85 (1993; Zbl 0779.53041)], by \textit{T. Akahori} and \textit{K. Miyajima}.