Foliations by Complex Leaves

Abstract: (Codimension-one) foliations by complex leaves are, roughly speaking, smooth partitions of a smooth manifold into complex manifolds of codimension-one (think of \Bbb C^n\times \Bbb R foliated by the level sets of the projection map onto \Bbb R). They are mixed objects, tangentially holomorphic and transversally smooth.

They appear in different situations. Examples include differentiable families of deformations of compact complex manifolds, Levi-flat hypersurfaces in complex projective space, every oriented 2-dimensional smooth foliation. But there exist also more exotic ones as a foliation of the 5-sphere by complex surfaces.

They can be seen as generalized deformations of complex manifolds, or as generalized complex structures on odd-dimensional smooth manifolds.

In this talk, I will explain the notion of foliations by complex leaves and describe (some of) the previous examples in detail. Then, taking the viewpoint of generalized deformations, I will compare classical deformations and foliated deformations in the particular example of Hopf manifolds.

This is a joint work with M. Nicolau and A. Verjovsky.

Refreshments will be served at 2:45 p.m. (Math Lounge, MATX 1115).