Lefschetz manifold

Definitions

If

${\displaystyle L_{[\omega ]}^{n-1}:H_{DR}^1(M)\to H_{DR}^{2n-1}}$

and

${\displaystyle L_{[\omega ]}^n:H_{DR}^0(M)\to H_{DR}^{2n}}$

are isomorphisms, then ${\displaystyle [\omega ]}$ is a Lefschetz element, or Lefschetz class. If

${\displaystyle L_{[\omega ]}^i:H_{DR}^{n-i}(M)\to H_{DR}^{n+i}(M)}$

is an isomorphism for all ${\displaystyle 0\le i\le n}$, then ${\displaystyle [\omega ]}$ is a strong Lefschetz element, or a strong Lefschetz class.

Let ${\displaystyle (M,\omega )}$ be a ${\displaystyle 2n}$-dimensional symplectic manifold. Then it is orientable, but maybe not compact. One says that ${\displaystyle (M,\omega )}$ is a Lefschetz manifold if ${\displaystyle [\omega ]}$ is a Lefschetz element, and ${\displaystyle (M,\omega )}$ is a strong Lefschetz manifold if ${\displaystyle [\omega ]}$ is a strong Lefschetz element.

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.

{Kähler manifolds} ${\displaystyle \subset }$ {strong Lefschetz manifolds} ${\displaystyle \subset }${Lefschetz manifolds} ${\displaystyle \subset }$ {symplectic manifolds}

Chal Benson and Carolyn S. Gordon proved in 1988 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 were given by Takumi Yamada in 2002.