Lagrangian submanifolds in Lefschetz pencils.

Francisco Presas
Stanford University

Given an almost contact structure, we select a fixed almost

complex structure on it and some additional geometric data. We show how

to construct sections of suitable sequences of complex bundles that are

"approximately C-R" in a suitable sense. We use these sections to

provide a number of geometric constructions in the manifold: existence

of submanifolds with special properties, existence of open book

decompositions, embeddings in the projective space preserving the

geometric structure, etc. As a corollary we show the existence of smooth

open book decompositions in a large class of 5-manifolds.

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