 
Summary: LEFSCHETZ PENCILS, BRANCHED COVERS AND
SYMPLECTIC INVARIANTS
DENIS AUROUX AND IVAN SMITH
Lectures given at the CIME summer school "Symplectic 4manifolds and algebraic
surfaces", Cetraro (Italy), September 210, 2003.
Two symplectic fibrations are never exactly the same. When you have
two fibrations, they might be canonically isomorphic, but when you look
closely, the points of one might be numbers while the points of the other
are bananas.
(P. Seidel, 9/9/03)
1. Introduction and background
This set of lectures aims to give an overview of Donaldson's theory of linear
systems on symplectic manifolds and the algebraic and geometric invariants to
which they give rise. After collecting some of the relevant background, we discuss
topological, algebraic and symplectic viewpoints on Lefschetz pencils and branched
covers of the projective plane. The later lectures discuss invariants obtained by
combining this theory with pseudoholomorphic curve methods.
1.1. Symplectic manifolds.
Definition 1.1. A symplectic structure on a smooth manifold M is a closed non
degenerate 2form , i.e. an element 2
