nLab hard Lefschetz theorem

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).

References

• Claire Voisin, section 6 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3

• Notes on Lefschetz decomposition pdf

• MO discussion Intuition for primitive cohomology

In the generality of matroids:

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

  [doi:10.4171/ICM2022/205, pdf, pdf]