Lefschetz Fibrations on Compact Stein Manifolds

TL;DR

This paper proves that every high-dimensional compact Stein manifold admits a Lefschetz fibration with Stein fibers and monodromy given by Dehn twists, extending known results from Stein surfaces to higher dimensions.

Contribution

It generalizes the existence of Lefschetz fibrations with specific monodromy to all compact Stein manifolds of dimension greater than four.

Findings

Every compact Stein manifold admits a Lefschetz fibration over the disk.

The monodromy is generated by right-handed Dehn twists along Lagrangian spheres.

The induced open book supports the boundary contact structure.

Abstract

Here we prove that up to diffeomorphism every compact Stein manifold W of dimension 2n+2>4 admits a Lefschetz fibration over the two-disk with Stein regular fibers, such that the monodromy of the fibration is a symplectomorphism induced by compositions of right-handed Dehn twists along embedded Lagrangian n-spheres on the generic fiber. This generalizes the Stein surface case of n=1, previously proven by Loi-Piergallini and Akbulut-Ozbagci. More precisely, we show that up to Liouville isomorphism any Weinstein domain W admits a compatible compact convex Lefschetz fibration with Weinstein regular fibers and with the same monodromy description stated above. Moreover, the induced convex open book supports the induced contact structure on the boundary of W.