For complex projective space with the FubiniStudy metric and associated Laplacede Rham operator $dd^\ast+d^\ast d$. How does one find a concrete description of the space of harmonic forms? That is, how does one find a basis of the space of forms $\omega$ for which $(dd^\ast+d^\ast d)(\omega)=0 $?
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Let $X$ be a compact Kahler manifold with Kahler form $\omega$. Then constant functions are obviously harmonic, and if $\alpha$ is harmonic then so is $\omega\wedge \alpha$ because this operation commutes with the Laplacian (see GriffithsHarris page 115). When $X=\mathbb{C}\mathbb{P}^n$, $\omega^i$is the sole harmonic form of degree $2i$, because it generates $H^{2i}(X)$. Footnotes


