Hypersurfaces symplectiques r\'eelles et pinceaux de Lefschetz r\'eels

Abstract

In a compact, symplectic real manifold, i.e supporting an antisymplectic involution, we use Donaldson's construction to build a codimension 2 symplectic submanifold invariant under the action of the involution. If the real part of the manifold is not empty, and if the symplectic form $\om$ is entire, then for all $k$ big enough, we can find a hypersurface Poincar\'e dual of $k[\omega]$ such that its real part has at least $k^{\dim X/4}$ connected components, up to a constant independant of $k$. Finally we extend to our real case Donaldson's construction of Lefschetz pencils.