TITLE: Intuition for Primitive Cohomology
QUESTION [32 upvotes]: In complex projective geometry, we have a specified Kähler class $\omega$ and we have a Lefschetz operator $L:H^i(X,\mathbb{C})\to H^{i+2}(X,\mathbb{C})$ given by $L(\eta)=\omega\wedge \eta$.  We then define primitive cohomology $P^{n-k}(X,\mathbb{C})=\ker(L^{k+1}:H^{n-k}(X)\to H^{n+k+2}(X))$, and we even have a nice theorem, the Lefschetz decomposition, that says $H^m(X,\mathbb{C})=\oplus_k L^kP^{n-2k}$.  Often, in papers, people just prove their result for primitive classes, as they seem to be easier to work with.
So, what exactly ARE primitive classes? Sure, they're things that some power of $\omega$ kills, but what's the intuition? Why are they an interesting distinguished class? Is there a good reason to expect this decomposition?

REPLY [16 votes]: If you take Poincare duals, then the wedge product becomes intersection, and the Kahler form $\omega$ becomes a hyperplane $H$. Then, the primitive cohomology in dimension $n - k$ consists exactly of those classes whose duals (which are homology classes in dimension $n + k$), when we intersect with $k + 1$ generic hyperplanes gives us zero. In other words, the homology classes which don't intersect with some $(n - k - 1)$-dimensional linear subspace of $\mathbb{CP}^n$. The Lefschetz theorem in this context just says that we can get all of the homology by taking those classes which don't intersect with a linear subspace of complementary dimension and then intersecting with hyperplanes.
In the case $k = 0$, we can choose the hyperplane to be the one at infinity, so the primitive homology consists of the $n$-dimensional homology classes which sit in the "$\mathbb{C}^n$" part of $\mathbb{CP}^n$, which I think is a pretty natural set of classes to look at.