nLab
hard Lefschetz theorem

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Context

Complex geometry

Symplectic geometry

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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 the Lefschetz operator

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) \,.

Using that ω\omega is a (1,1)(1,1)-form this means equivalently in terms of Dolbeault cohomology that for all (p+q)d(p+q) \leq d we have an isomorphism

L d(p+q):H p,q(X)H dq,dp(X). 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 XX.

References

Created on June 10, 2014 at 03:33:17. See the history of this page for a list of all contributions to it.