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The Kodaira embedding theorem yields as a corollary that a compact Kähler manifold $X$ with $h^{2,0} =0$ is projective.

Is there a weaker relation on Hodge numbers that implies that a compact Kähler manifold is projective?

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    $\begingroup$ You can't say much from the Hodge numbers $h^{pq}$ alone. Eg there are non-projective K3 surfaces (with $h^{2,0}=1$) and non-projective abelian surfaces (with $h^{2,0}=2$). $\endgroup$
    – Ennio Mori cone
    Commented Nov 5, 2021 at 11:35

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