Lemma 11.1.3.BF56C7366FC6 For a smooth real $(1,1)$-form $\theta $ and an integer $q\geq 1$,
\[ \mathrm{dd}^{\mathrm{c}}(\theta ^ q) =q\, \theta ^{q-1}\wedge \mathrm{dd}^{\mathrm{c}}\theta +q(q-1)\, \theta ^{q-2}\wedge \mathrm{d}\theta \wedge \mathrm d^{\mathrm c}\theta , \]
where the second term is omitted when $q=1$.
Proof.
For $q=1$ there is nothing to prove. For $q\geq 2$, since $\theta $ has even degree, $\mathrm d^{\mathrm c}(\theta ^ q)=q\, \theta ^{q-1}\wedge \mathrm d^{\mathrm c}\theta $. Applying $\mathrm{d}$ and using again that $\theta ^{q-1}$ has even degree, we obtain
\[ \mathrm{d}\mathrm d^{\mathrm c}(\theta ^ q) =q(q-1)\, \theta ^{q-2}\wedge \mathrm{d}\theta \wedge \mathrm d^{\mathrm c}\theta +q\, \theta ^{q-1}\wedge \mathrm{d}\mathrm d^{\mathrm c}\theta . \]
$\square$
Pome
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