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)?