nLab Lefschetz decomposition

Idea

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.

Definition

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

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

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

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

Define the primitive cohomology of $X$ in degree $d-k$ to be the kernel

$$P^{d-k}(X) \coloneqq ker\left( H^{d-k}(X) \stackrel{L^{k+1}}{\longrightarrow} H^{d+k+2}(X) \right) \,.$$

The hard Lefschetz theorem then implies the follows isomorphism, which is the Lefschetz decomposition

$$H^{d}(X,\mathbb{C}) \simeq \underset{k}{\oplus} L^k P^{d-2k}(X) \,.$$

Related concepts

• Hodge structure

• hard Lefschetz theorem

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