geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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
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
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
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).
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:
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