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
On a Kähler manifold $(X,\omega)$, the Lefschetz decomposition is a decomposition of the de Rham cohomology into differential forms which are annihilted by wedge product with some power of the Kähler form $\omega$ times some other power of the 2-form.
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 a function
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
Define the primitive cohomology of $X$ in degree $d-k$ to be the kernel
The hard Lefschetz theorem then implies the follows isomorphism, which is the Lefschetz decomposition
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
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