Stein manifolds in symplectic geometry
Emmanuel Giroux
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 geometry.