Lefschetz manifold

In mathematics, a Lefschetz manifold is a particular kind of symplectic manifold ${\displaystyle (M^{2n},\omega )}$, sharing a certain cohomological property with Kähler manifolds, that of satisfying the conclusion of the Hard Lefschetz theorem. More precisely, the strong Lefschetz property asks that for ${\displaystyle k\in \{1,\ldots ,n\}}$, the cup product

${\displaystyle \cup [\omega ^{k}]\colon H^{n-k}(M,\mathbb {R} )\to H^{n+k}(M,\mathbb {R} )}$

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.[1]

Lefschetz maps

Let ${\displaystyle M}$ be a (${\displaystyle 2n}$)-dimensional smooth manifold. Each element

${\displaystyle [\omega ]\in H_{DR}^{2}(M)}$

of the second de Rham cohomology space of ${\displaystyle M}$ induces a map

${\displaystyle L_{[\omega ]}:H_{DR}(M)\to H_{DR}(M),[\alpha ]\mapsto [\omega \wedge \alpha ]}$

called the Lefschetz map of ${\displaystyle [\omega ]}$. Letting ${\displaystyle L_{[\omega ]}^{i}}$ be the ${\displaystyle i}$th iteration of ${\displaystyle L_{[\omega ]}}$, we have for each ${\displaystyle 0\leq i\leq n}$ a map

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

If ${\displaystyle M}$ is compact and oriented, then Poincaré duality tells us that ${\displaystyle H_{DR}^{n-i}(M)}$ and ${\displaystyle H_{DR}^{n+i}(M)}$ are vector spaces of the same dimension, so in these cases it is natural to ask whether or not the various iterations of Lefschetz maps are isomorphisms.

The Hard Lefschetz theorem states that this is the case for the symplectic form of a compact Kähler 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 ${\displaystyle 0\leq i\leq n}$, 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[2] 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.[3]

Notes

1. Thurston, William P. (1976). "Some simple examples of symplectic manifolds". Proceedings of the American Mathematical Society. 55 (2): 467. doi:10.2307/2041749. JSTOR 2041749. MR 0402764.
2. Benson, Chal; Gordon, Carolyn S. (1988). "Kähler and symplectic structures on nilmanifolds". Topology. 27 (4): 513–518. doi:10.1016/0040-9383(88)90029-8. MR 0976592.
3. Yamada, Takumi (2002). "Examples of compact Lefschetz solvmanifolds". Tokyo Journal of Mathematics. 25 (2): 261–283. doi:10.3836/tjm/1244208853. MR 1948664.