Question

For complex projective space with the Fubini-Study metric and associated Laplace-de 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 $?

Answer 1

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 Griffiths-Harris 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

1. Of course, I meant that $\omega^i$ is a basis for degree $2i$ harmonic forms.
2. There are no harmonic forms of odd degree, because there is no cohomology.
3. Curiously, this argument is valid for any Kahler metric on projective space, not just the Fubini-Study metric.
4. We can argue differently by noting the Fubini-Study metric is invariant under the action of $U(n+1)$, so the same is true for harmonic forms...