Summary: LAGRANGIAN SUBMANIFOLDS AND LEFSCHETZ PENCILS.
DENIS AUROUX, VICENTE MU~NOZ, AND FRANCISCO PRESAS
Abstract. Given a Lagrangian submanifold in a symplectic manifold and a Morse function
on the submanifold, we show that there is an isotopic Morse function and a symplectic
Lefschetz pencil on the manifold extending the Morse function to the whole manifold. From
this construction we define a sequence of symplectic invariants classifying the isotopy classes
of Lagrangian spheres in a symplectic 4-manifold.
For a symplectic manifold (M, ), S. Donaldson has proved in  the existence of symplectic
Lefschetz pencils using the recently introduced asymptotically holomorphic techniques [8, 2].
To give the definition of a Lefschetz pencil, recall that a chart (, U), = (z1, . . . , zn) :
U M Cn is adapted at a point p M if (p) = 0 and (J0) is tamed by (where J0
is the standard complex structure on Cn); equivalently, this means that complex lines in the
local coordinates are symplectic with respect to .
Definition 1.1. A symplectic Lefschetz pencil associated to a symplectic manifold (M, )
consists of the following data:
(i) A codimension 4 symplectic submanifold N.
(ii) A surjective map : M - N CP1
(iii) A finite set of points M - N away from which the map is a submersion.