Take the 2-minute tour ×  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?  share|improve this
question Stupid notational confusion: by $L^{k+1}$ do you mean L
iterated k+1 times or L applied to $H^{k+1}$? –  Ilya Grigoriev
Feb 8 '10 at 18:03 Sorry, meant applied $k+1$ times. I'll edit.
–  Charles Siegel Feb 8 '10 at 18:04 Typ