PomeProposition 11.2.8.E345C8B2A48C If $\mathrm d\theta =0$, then $DE_{\theta ,u}(v)=\int _ Xv\theta _ u^ n$ for every pair of smooth real functions $u,v$.
11.2.6 Radial energy and closed backgroundsB9357DC44463
Integrating the Monge–Ampère one-form along the segment from $0$ to $u$ defines a functional for an arbitrary background. Let $\theta $ be a smooth real $(1,1)$-form on $X$. For $u\in C^\infty (X,\mathbb R)$, put $\theta _ u=\theta +\mathrm{dd}^{\mathrm{c}}u$ and define the radial energy
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11.2.7
\begin{equation} \begin{aligned} E_\theta (u) & =\int _0^1\int _ Xu(\theta +t\mathrm{dd}^{\mathrm{c}}u)^ n\, \mathrm{d}t\\ & =\frac1{n+1}\sum _{j=0}^ n \int _ Xu\, \theta _ u^ j\wedge \theta ^{n-j}. \end{aligned} \end{equation}
To obtain the second expression, write $\theta +t\mathrm{dd}^{\mathrm{c}}u=(1-t)\theta +t\theta _ u$, expand, and use
\[ \binom nj\int _0^1t^ j(1-t)^{n-j}\, \mathrm dt=\frac1{n+1}. \]
We write $DE_{\theta ,u}(v)$ for the derivative of $E_\theta $ at $u$ in the direction $v$. When $\theta $ is closed, integration by parts makes the derivatives of the mixed terms telescope.
Proof. Differentiate the finite sum in (11.2.7). The term indexed by $j$ contributes
\[ \int _ Xv\theta _ u^ j\wedge \theta ^{n-j} +j\int _ Xu\mathrm{dd}^{\mathrm{c}}v\wedge \theta _ u^{j-1}\wedge \theta ^{n-j}, \]
the second summand being absent for $j=0$. For $j\geq 1$, the form $\theta _ u^{j-1}\wedge \theta ^{n-j}$ is closed, so by lemma 11.2.2 we may interchange $u$ and $v$ in the second summand. Using $\mathrm{dd}^{\mathrm{c}}u=\theta _ u-\theta $, we can then write the contribution of the term with $j\geq 1$ as
\[ (j+1)\int _ Xv\theta _ u^ j\wedge \theta ^{n-j} -j\int _ Xv\theta _ u^{j-1}\wedge \theta ^{n-j+1}. \]
Together with the contribution $\int _ Xv\theta ^ n$ of the term $j=0$, these expressions telescope to $(n+1)\int _ Xv\theta _ u^ n$. Dividing by $n+1$ completes the proof. $\square$
Thus, when $\mathrm d\omega =0$, the restriction of $E_\omega $ to $\mathcal H_\omega $ is the primitive required in theorem 11.2.4.
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