Kähler class

Kähler forms in the same class differ by an exact real $2$-form. On a compact Kähler manifold, the $\partial\bar\partial$-lemma refines this: two cohomologous Kähler forms differ by $i\partial\bar\partial\varphi$ for a global real smooth function $\varphi$, up to the chosen normalization. Positivity is then the open inequality that the modified form must continue to satisfy.

If $X$ has complex dimension $n$ and is compact, then

so $[\omega]^n\neq0$. More generally, every power $[\omega]^k$ is nonzero for $0\leq k\leq n$, as shown in Demailly, Chapter VI, §4, Consequence 4.3.

On complex projective space, the Fubini–Study form represents a Kähler class; with the standard integral normalization, it generates $H^2(\mathbb{CP}^n;\mathbb Z)$. By contrast, the zero class on a positive-dimensional compact complex manifold is not Kähler: if $\omega=d\eta$, then Stokes' theorem would give $\int_X\omega^n=0$, contradicting positivity. Demailly records both the Fubini–Study class and its normalization in Chapter VI, §4, Example 4.4.

1. Jean-Pierre Demailly, Complex Analytic and Differential Geometry, 2012. Author-hosted text. Relevant: Chapter VI, §4, especially Consequence 4.3 and Example 4.4.