PomeCorollary 11.1.18.9578EA44189A The inequality (11.1.17) does not hold for every Hermitian form on a compact complex manifold.
Corollary 11.1.189578EA44189A
Proof. Let $X$, $\omega $, $u_ t$ and $B=\int _ X\beta _1\wedge \beta _2$ be as in example 11.1.12. For $-\pi ^{-2}<t<0$, set $u=0$ and $v=u_ t+C$ with $C>|t|$. Then $\{ u<v\} =X$, but
\[ \int _ X\omega _ v^2=(4-\pi ^2t)B>4B=\int _ X\omega _ u^2. \]
Thus this pair violates (11.1.17). $\square$
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