Stein manifolds in symplectic geometry
In complex analysis, stein manifolds are defined as complex manifolds admitting
"many" holomorphic maps to the complex line. According to Grauert's solution of
the Levi problem, they are characterized by a certain pseudoconvexity property
whose symplectic nature has been first pointed out by Eliashberg. Eliashberg
also proved that the existence of a Stein structure on a smooth manifold of
sufficiently high dimension is governed by simple topological obstructions. In
the last decade, the importance of Stein manifolds in symplectic geometry has
been emphasized by Donaldson's theory of symplectic hyperplane sections and the
existence of supporting open books for contact structures.
After reviewing this context, we will show how Donaldson's construction of
symplectic Lefschetz pencils on closed integral symplectic manifolds can be
adapted to obtain, on any Stein domain, Lefschetz fibrations over the disc which
are asymptotically unique up to stabilization by positive Lagrangian plumbings.
as an application, we will prove taht a contact manifold is Stein fillable if
and only if it contains a supporting open book whose monodromy is a product of
right-handed symplectic Dehn twists along Lagrangian spheres.
Finally, we will discuss recent results of Eliashberg and Muyrphy exhibiting a
speial cllas of very flexible Stein structures, and see how such structures
might be useful to attack a longstanding open problem in contact