PomeLemma 11.1.2.3B0A8DD950AD For a smooth real function $f$ and a smooth form $T$ of bidegree $(n-1,n-1)$,
\[ \int _ X\mathrm{dd}^{\mathrm{c}}f\wedge T=\int _ X f\mathrm{dd}^{\mathrm{c}}T. \]
11.1.1 Integration by parts with a nonclosed formFD6E63098594
Integration by parts moves the complex Hessian of a function onto the form against which it is integrated. When that form is a power of a nonclosed $(1,1)$-form, its complex Hessian contains, besides the complex Hessian of the form itself, a quadratic term in its exterior derivative.
Proof. Since $\mathrm d^{\mathrm c}f$ is a one-form, $\mathrm{d}(\mathrm d^{\mathrm c}f\wedge T)=\mathrm{dd}^{\mathrm{c}}f\wedge T-\mathrm d^{\mathrm c}f\wedge \mathrm{d}T$, and Stokes' theorem gives the first equality in
\[ \int _ X\mathrm{dd}^{\mathrm{c}}f\wedge T =\int _ X\mathrm d^{\mathrm c}f\wedge \mathrm{d}T =\int _ X f\mathrm{dd}^{\mathrm{c}}T. \]
For the second, recall that the integral of $\mathrm d^{\mathrm c}\sigma $ vanishes for every form $\sigma $ of total degree $2n-1$, by Stokes' theorem for $\partial $ and $\bar\partial $. We apply this to $\sigma =f\, \mathrm{d}T$ and use $\mathrm d^{\mathrm c}\mathrm{d}=-\mathrm{d}\mathrm d^{\mathrm c}$. $\square$
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
\[ \mathrm{d}\mathrm d^{\mathrm c}(\theta ^ q) =q(q-1)\, \theta ^{q-2}\wedge \mathrm{d}\theta \wedge \mathrm d^{\mathrm c}\theta +q\, \theta ^{q-1}\wedge \mathrm{d}\mathrm d^{\mathrm c}\theta . \]
$\square$
Thus the quadratic torsion term $\mathrm{d}\omega \wedge \mathrm d^{\mathrm c}\omega $ can make the higher powers of $\omega $ fail to be $\mathrm{dd}^{\mathrm{c}}$-closed even when $\mathrm{dd}^{\mathrm{c}}\omega =0$. The mass expansion below involves the complex Hessians of all these powers separately.
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