Transversely holomorphic foliations and CR structures (Q2713040)

Let \(M\) be a smooth manifold. A subbundle \(V\) of the complexified tangent bundle \(C\otimes TM\) of \(M\) is called involutive if \([V,V]\subset V\). An involutive bundle \(V\) is called an elliptic structure (resp. CR structure) if \(V+\overline V = C\otimes TM\) and \(d = \text{rank}(V\cap \overline V) > 0\) (resp. \(V + \overline V\neq C\otimes TM\) and \(V\cap \overline V = \{0\}\)). An elliptic structure \((M,V)\) is called to dominate a CR structure \((M,V_0)\) if \(V_0\subset \overline V\) and \(\text{rank}_{\mathbb{C}}V = \text{rank}_{\mathbb{C}}V_0+ d\) holds.NEWLINENEWLINENEWLINEThe author proves that an elliptic structure \((M,V)\) dominates some \(C^\omega\)CR structure \((M,V_0)\) if and only if the foliation associated to \((M, V)\) is complexifiable in the sense of \textit{A. Haefliger} and \textit{D. Sundararaman} [Math. Ann. 272, 23-27 (1985; Zbl 0562.57013)].