Lefschetz Fibrations on Compact Stein Manifolds

Here we prove that up to diffeomorphism every compact Stein manifold W of dimension 2n + 2 > 4 admits a Lefschetz fibration over the disk D 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 AkbulutOzbagci. 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 boundary convex open book supports the induced contact structure on ∂W .