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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 $P^{n-k}(X,\mathbb{C})=\ker(L^{k+1}:H^{n-k}(X)\to H^{n+k}(X))$, 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?

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?

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 $P^{n-k}(X,\mathbb{C})=\ker(L^{k+1})$, 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?

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 $P^{n-k}(X,\mathbb{C})=\ker(L^{k+1}:H^{n-k}(X)\to H^{n+k}(X))$, 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?