Primitive Cohomology Useful?

Question

In her book, after proving the hodge decomposition, Voisin spends time discussing primitive cohomology $H^r(X, \mathbb{C})_{prim} = \ker L^{n-r+1} \subset H^r(X, \mathbb{C})$ (where $L$ is the Lefschetz operator). She proves several general theorems regarding/using primitive cohomology (Hodge index, Lefschetz decomposition, a bilinear form on $H^k(X,\mathbb{C})$ behaving in a controlled way on primitive cohomology) and establishes some technical results (if $\omega$ is a primitive form then there is a formula for $*\omega$ in terms of the Lefschetz operator and $\omega$).

$\textbf{Question: }$ I'm having a hard time understanding why one should care about primitive cohomology. Can you deduce lots of interesting facts about nonsingular complex projective varieties with say the Lefschetz decomposition as was the case with the Hodge decomposition? What are some typical applications? I'd really like some examples to illustrate if/how primitive cohomology is useful.

Specifically, I am interested in how primitive cohomology could be useful on a "daily basis". For example, let $X, Y$ be smooth complex projective varieties. Sometimes one can deduce that there are no surjective maps $X \xrightarrow{\phi} Y$ because such maps induce injective maps on cohomology (which preserve Hodge structure). Can primitive cohomology give a more refined obstruction to the existence of $\phi$ in certain cases?

$\textbf{Computing}$

1.) How about primitive cohomology? This depends on the choice of a Kahler form. Do the dimensions of the primitive cohomology groups not depend on the choice of Kahler form? It's not clear to me if primitive cohomology of abelian varieties depends only on the dimension. Does one know the dimensions of primitive cohomology groups of an abelian variety? How about other classes of varieties? For a K3 surface, it seems like one can give the dimensions of primitive cohomology groups independent of kahler form, the main point is $h^{1,1}$, where primitive cohomology has dimension 19.

$\textbf{Functoriality}$

2.) A surjective map of smooth complex projective varieties is injective on cohomology and a map of hodge structures. A finite surjective map pulls back ample divisiors to ample divisors, so if we choose kahler classes appropriately, then such a map induces a map on primitive cohomology. Does a more general class of maps induce maps on primitive cohomology (if we choose kahler classes appropriately)?

Answer 1

Let me work in homology, which is closer to the way someone like Lefschetz would have thought about it. Given a smooth complex projective variety in $X\subset \mathbb{P}^N$, we would like to understand the homology inductively. So take a general hyperplane $H\subset \mathbb{P}^N$, then $Y=X\cap H$ is again smooth (Bertini) of smaller dimension. The Lefschetz hyperplane theorem says that $H_i(Y)\to H_i(X)$ is surjective when $i<\dim X$. So in this range, all of the homology is effectively captured by $Y$. But when $i=\dim X$, the simple minded induction breaks down, because there will be a kernel which would be the primitive homology in the middle degree. This is the part that is genuinely new, and that needs to be understood on its own terms. This perhaps the simplest answer.

Switching to cohomology, the hard Lefschetz theorem shows that cohomology decomposes into primitive parts, so these are the essential constituents for cohomology. The hard Lefschetz theorem has an enormous number of implications for the topology of smooth projective varieties or more generally compact Kahler manifolds. Dan's answer gives one such application. Here is another: the Betti numbers of smooth projective variety satisfy $b_0\le b_2\le b_4\ldots$ and $b_1\le b_3\le\ldots$ up to the dimension, after which they decrease. Added in response to the follow up question: I don't have the time or energy to create a new answer, so I'll comment here. The successive differences $b_3-b_1$ etc. give the dimensions of primitive cohomology.

Answer 2

One application is Deligne's theorem on the degeneration of the Leray spectral sequence. Strictly speaking this is not an interesting fact about smooth projective varieties but about families of smooth projective varieties, and their relative cohomology $\mathrm R^if_\ast \mathbf Q$.

More precisely, the result is that for $f \colon X \to S$ a smooth and projective morphism of $\mathbf C$-varieties, there is an isomorphism $$ \mathrm Rf_\ast\mathbf Q \cong \bigoplus_i \mathrm R^i f_\ast\mathbf Q[-i]$$ in the derived category of $S$. In particular, the Leray spectral sequence $E_2^{pq} = H^p(S,\mathrm R^q f_\ast \mathbf Q) \implies H^{p+q}(X)$ degenerates, but the derived category result is "universal".

The proof is very formal and the key idea is the existence of a Lefschetz operator and a decomposition into primitive parts on the relative cohomology. Deligne's paper is very readable: Théorème de Lefschetz et critères de dégénérescence de suites spectrales

Answer 3

One more application: the singular cohomology functor (with coefficients in a field) restricted to smooth projective complex varieties factorizes through the semi-simple category of polarizable pure Hodge structures. There is a certain (somewhat complicated) extension of this result to cohomology of all complex varieties.