# nLab hard Lefschetz theorem

Contents

complex geometry

### Examples

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

cohomology

# Contents

## Idea

For $(X,\omega)$ a Kähler manifold the operation of forming the wedge product with the symplectic form $\omega \in \Omega^{1,1}(X)$ induces on de Rham cohomology the Lefschetz operator

$L \;\colon\; H^k(X) \longrightarrow H^{k+2}(X) \,.$

The hard Lefschetz theorem asserts that if $X$ is compact with complex dimension $dim_{\mathbb{C}}(X)= d$, then for all $k \geq 0$ the $k$th power of the $L$-operation induces an isomorphism

$L^k \;\colon\; H^{d-k}(X) \stackrel{\simeq}{\longrightarrow} H^{d+k}(X) \,.$

Using that $\omega$ is a $(1,1)$-form this means equivalently in terms of Dolbeault cohomology that for all $(p+q) \leq d$ we have an isomorphism

$L^{d-(p+q)} \;\colon\; H^{p,q}(X) \stackrel{\simeq}{\longrightarrow} H^{d-q,d-p}(X) \,.$

This exhibits the symmetry of the Hodge diamond? under reflection about the horizontal diagonal.

The hard Lefschetz theorem induces the Lefschetz decomposition (see there) of the de Rham cohomology of $X$.

The hard Leftschetz theorem also applies to other objects, for instance, to matroids (see Huh 22).

In the generality of matroids:

• June Huh, Combinatorics and Hodge theory, Proc. Int. Cong. Math. 1 (2022)