= Odd Betti numbers of a compact Kähler manifold are even
The <Kähler Laplacian identity> makes the space of complex <harmonic differential forms> of degree $r$ the direct sum of its $(p,q)$ parts with $p+q=r$. Conjugation identifies the $(p,q)$ and $(q,p)$ harmonic spaces. If $r$ is odd these are distinct and pair off, so the <Betti number> in degree $r$ is even. This is a topological obstruction to existence of a <Kähler metric>, exhibited by a <Hopf surface> with first Betti number one.
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