Let $(X,\omega)$ be a Kahler manifold of dimension $n$, let $L = \omega \wedge -$ and let $\Lambda$ denote its adjoint. There is a unique ''primitive decomposition'' of a $k$-form $u$ which looks like $$ u = \sum_{r \geq \max(k-n,0)} L^r u_r $$ for some $(k-2r)$-forms $u_r$ such that $\Lambda u = 0$. This is proved in many places, but there are not many examples or calculations that I've seen!

For example, if I write down some form $\eta$ on $\mathbb{C}^n$ (which is a Kahler manifold when equipped with the Kahler form $\omega = \frac{i}{2} \sum_{j=1}^n dz_j \wedge d\overline{z}_j$), then how would I find the primitive decomposition of $\eta$?

Is there some algorithmic way, in general, to calculate the primitive decomposition of a form?

  • $\begingroup$ Can you find forms representing the cohomology of $\mathbb{CP}^n$? $\endgroup$ – 54321user May 3 '17 at 18:05

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