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 $?
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2$\begingroup$ Since the cohomology of $\mathbb{C}\mathbb{P}^n$ is so simple, I think the harmonic forms would just be multiples of the Kahler form and the constant functions. $\endgroup$– Donu ArapuraMay 7, 2011 at 15:22
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1 Answer
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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 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
- Of course, I meant that $\omega^i$ is a basis for degree $2i$ harmonic forms.
- There are no harmonic forms of odd degree, because there is no cohomology.
- Curiously, this argument is valid for any Kahler metric on projective space, not just the Fubini-Study metric.
- 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...
answered May 7, 2011 at 15:44
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$\begingroup$ Does anyone know how to do an umlaut (and similar diacritics) on Mathoverflow? Writing K\"ahler doesn't seem to work. $\endgroup$ May 8, 2011 at 16:35