Primitive cohomology

Let $X$ be a compact Kähler manifold of complex dimension $n$ , and let $L$ be the Lefschetz operator determined by its Kähler class. For $0\leq k\leq n$ , the primitive cohomology in degree $k$ is

P^k(X,\mathbb C) =\ker\!\left(L^{\,n-k+1}:H^k(X,\mathbb C)\longrightarrow H^{2n-k+2}(X,\mathbb C)\right).

Its component of bidegree $(p,q)$ , where $p+q=k$ , is $P^{p,q}(X)=P^k(X,\mathbb C)\cap H^{p,q}(X)$ . Thus primitiveness is relative to the chosen Kähler class; it is not merely a property of the underlying cohomology group. Higher-degree classes are described by Lefschetz powers of primitive classes in degrees at most $n$ .

Equivalent harmonic description

Choose the Kähler metric determined by the Kähler form, and represent a class by its unique harmonic form. The class is primitive exactly when this representative is annihilated by the adjoint Lefschetz operator $\Lambda$ . This identifies the cohomological kernel above with the pointwise notion of a primitive harmonic form.

Lefschetz decomposition

The hard Lefschetz theorem yields the direct-sum decomposition

H^k(X,\mathbb C) =\bigoplus_{r\geq \max(0,k-n)}L^rP^{k-2r}(X,\mathbb C).

Every cohomology class is therefore uniquely assembled from primitive classes and powers of the Kähler class. The decomposition respects bidegree because $L$ has type $(1,1)$ .

Examples and scope

On projective space $\mathbb{CP}^n$ , the only primitive cohomology is $P^0$ : every positive even-degree generator is a positive power of the Kähler class. By contrast, the middle cohomology of a projective hypersurface can contain a substantial primitive summand.

References

Claire Voisin, Hodge Theory and Complex Algebraic Geometry I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002. Publisher record. Relevant: §6.2, especially Proposition 6.24 and Theorem 6.25.
Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005. Publisher record. Relevant: Chapter 3, the Lefschetz decomposition and primitive cohomology.