nLab
Lefschetz decomposition

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Context

Complex geometry

Symplectic geometry

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

On a Kähler manifold (X,ω)(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 ω\omega times some other power of the 2-form.

Definition

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

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

The hard Lefschetz theorem asserts that if XX is compact with complex dimension dim (X)=ddim_{\mathbb{C}}(X)= d, then for all k0k \geq 0 the kkth power of the LL-operation induces an isomorphism

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

Define the primitive cohomology of XX in degree dkd-k to be the kernel

P dk(X)ker(H dk(X)L k+1H d+k+2(X)). 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,)kL kP d2k(X). H^{d}(X,\mathbb{C}) \simeq \underset{k}{\oplus} L^k P^{d-2k}(X) \,.

References

Last revised on September 17, 2017 at 16:17:19. See the history of this page for a list of all contributions to it.