In mathematics, a Lefschetz manifold is a particular kind of symplectic manifold
Contents
be an isomorphism.
The topology of these symplectic manifolds is severely constrained, for example their odd Betti numbers are even. This remark leads to numerous examples of symplectic manifolds which are not Kähler, the first historical example is due to William Thurston.
Lefschetz maps
Let
of the second de Rham cohomology space of
called the Lefschetz map of
If
The Hard Lefschetz theorem states that this is the case for the symplectic form of a compact Kähler manifold.
Definitions
If
and
are isomorphisms, then
is an isomorphism for all
Let
Where to find Lefschetz manifolds
The real manifold underlying any Kähler manifold is a symplectic manifold. The strong Lefschetz theorem tells us that it is also a strong Lefschetz manifold, and hence a Lefschetz manifold. Therefore we have the following chain of inclusions.
In, Chal Benson and Carolyn S. Gordon proved that if a compact nilmanifold is a Lefschetz manifold, then it is diffeomorphic to a torus. The fact that there are nilmanifolds that are not diffeomorphic to a torus shows that there is some space between Kähler manifolds and symplectic manifolds, but the class of nilmanifolds fails to show any differences between Kähler manifolds, Lefschetz manifolds, and strong Lefschetz manifolds.
Gordan and Benson conjectured that if a compact complete solvmanifold admits a Kähler structure, then it is diffeomorphic to a torus. This has been proved. Furthermore, many examples have been found of solvmanifolds that are strong Lefschetz but not Kähler, and solvmanifolds that are Lefschetz but not strong Lefschetz. Such examples can be found in.