On the complex geometry of a class of non-Kählerian manifolds (Q1293992)

In this paper, complex, compact, non-Kählerian manifolds are studied. The authors presented [in Math. Ann. 306, 781-817 (1996; Zbl 0860.32001)] a procedure to build complex structures on a product of two spheres, in particular on elliptic curves, Hopf manifolds, Calabi-Eckmann manifolds. Using vector fields of Seigel type, \textit{S. Lopez de Medrano} and \textit{A. Verjovski} [Bol. Soc. Bras. Mat., Nova Ser. 28, 253-269 (1997; Zbl 0901.53021)] built a large class of compact complex manifolds (called LM-V manifolds for short). Except in the case of elliptic curves, LM-V manifolds do not admit a Kähler structure (in fact there is no symplectic structure). It is shown in this article that these manifolds enjoy a very special property: on any such manifold there exists a naturally defined holomorphic vector field \(\eta \) which is non-singular and transversely Kählerian. (For a complex manifold \(M\) and \(v\) a real non-singular vector field on \(M\), a real 2-form \(\omega \) is said to be transversely Hermitian with respect to \(v\) if 1. \(J\omega = \omega \) and, for any \( z \in M\), \(\text{Ker }\omega _z = {\mathbb C}v_z = {\mathbb R}v_z\oplus {\mathbb R}Jv_z\), and 2. the Hermitian quadratic form \(h\) on \(TM_{{\mid }{\mathbb C}v}\) induced by \(\omega \) is positively definite, where \(h\) is given by \(h(u_1, u_2) = \omega (Ju_1, u_2) + i\omega (u_1, u_2).\) If the 2-form \(\omega \) is closed, we say that \(v\) is transversely Kählerian.) So, the foliation defined by \(\eta \) is transversely modelled on a Kählerian manifold. This generalizes the fact that classical Hopf manifolds and Calabi-Eckmann manifolds are principal bundles over Kählerian manifolds (namely \({\mathbb P}^n\) and \({\mathbb P}^r\times {\mathbb P}^s\)) with fibre an elliptic curve. In Theorem 1 it is proved that analytic subsets of appropriate dimensions are tangent to this vector field \(\eta\), extending the property known for classical Hopf manifolds and Calabi-Eckmann manifolds to the case of LM-V manifolds. Theorem 2 gives a precise description, in the generic case, of these sets, so of any complex submanifold of a certain dimension, of the LM-V manifolds. In the proof, an important role is played by some complex abelian groups which are biholomorphic to big domains in LM-V manifolds.