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.

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?