PomeLemma 11.2.2.8A9C3139B5B5 For a smooth real $(n-1,n-1)$-form $\Omega $, the pairing $B(v,w)=\int _ Xw\mathrm{dd}^{\mathrm{c}}v\wedge \Omega $ is symmetric on smooth real functions if and only if $\mathrm{d}\Omega =0$.
11.2.1 Primitives of the Monge–Ampère one-form49D0776D0F6B
On $\mathcal H_\omega $ consider the Monge–Ampère one-form
\[ \alpha _ u(v)=\int _ Xv\, \omega _ u^ n,\qquad v\in C^\infty (X,\mathbb R). \]
Let $U\subseteq \mathcal H_\omega $ be open in the $C^2$ topology on smooth functions. A primitive of $\alpha $ on $U$ is a functional $E$ on $U$ such that, for every $u\in U$ and every smooth real direction $v$, the directional derivative
\[ DE_ u(v)=\left.\frac{\mathrm d}{\mathrm dt}\right|_{t=0}E(u+tv) \]
exists and equals $\alpha _ u(v)$. Restrict $E$ to the intersection of $U$ with a finite-dimensional affine plane. Its first derivatives are values of $\alpha $, which depend smoothly on the parameters, so its mixed second derivatives agree. The following integration-by-parts criterion identifies the differential condition imposed by this symmetry.
Proof. By lemma 11.1.2, $B(v,w)=\int _ Xv\mathrm{dd}^{\mathrm{c}}(w\Omega )$. Since $v$ is arbitrary, symmetry is therefore equivalent to the pointwise identity
\[ \mathrm{dd}^{\mathrm{c}}(w\Omega )-\mathrm{dd}^{\mathrm{c}}w\wedge \Omega =w\mathrm{dd}^{\mathrm{c}}\Omega +\mathrm{d}w\wedge \mathrm d^{\mathrm c}\Omega -\mathrm d^{\mathrm c}w\wedge \mathrm{d}\Omega =0 \]
for every smooth real $w$. Taking $w=1$ gives $\mathrm{dd}^{\mathrm{c}}\Omega =0$. The remaining expression is linear in the first jet of $w$, so we may complexify the test functions. At a chosen point, take a holomorphic linear function multiplied by a cutoff function, so that $\partial w=\lambda $ is an arbitrary $(1,0)$-covector and $\bar\partial w=0$ at that point. There the first-order expression equals
\[ \frac{\mathrm i}{2\pi }\bigl (\partial w\wedge \bar\partial \Omega -\bar\partial w\wedge \partial \Omega \bigr ). \]
It follows that $\lambda \wedge \bar\partial \Omega =0$ for every $(1,0)$-covector $\lambda $. Since the wedge pairing between $(1,0)$-covectors and forms of bidegree $(n-1,n)$ is nondegenerate, $\bar\partial \Omega =0$, and $\partial \Omega =0$ because $\Omega $ is real. Conversely, if $\mathrm{d}\Omega =0$, then $\partial \Omega =\bar\partial \Omega =0$ by type, so $\mathrm d^{\mathrm c}\Omega =0$, and the same product formula proves symmetry. $\square$
Lemma 11.2.3.8535B6F80ED4 Let $T$ be a form of type $(2,1)+(1,2)$ on a complex vector space of dimension $n\geq 2$. If $T\wedge A^{n-2}=0$ for every real $(1,1)$-form $A$ in an open neighborhood of a positive form, then $T=0$.
Proof. For $n=2$ the assertion is immediate. Otherwise the expression $T\wedge A^{n-2}$ is polynomial in the real coefficients of $A$, so it vanishes for every real $(1,1)$-form $A$. Polarization and then complexification give $T\wedge A_1\wedge \cdots \wedge A_{n-2}=0$ for arbitrary complex $(1,1)$-forms $A_ j$, and such products span all forms of bidegree $(n-2,n-2)$.
Since these wedges preserve type, the identity holds separately for the two type components of $T$. Wedging the identity for $T^{2,1}$ with arbitrary $(0,1)$-covectors, we obtain test forms spanning bidegree $(n-2,n-1)$, which pairs perfectly with bidegree $(2,1)$ into top degree. Thus $T^{2,1}=0$. The component $T^{1,2}$ is treated in the same way with $(1,0)$-covectors. $\square$
Theorem 11.2.4.7D069B347672 For $n\geq 2$, the one-form $\alpha $ has a primitive on $\mathcal H_\omega $ if and only if $\mathrm{d}\omega =0$. In fact, a primitive on a $C^2$-open neighborhood of $0$ already forces $\mathrm{d}\omega =0$.
Proof. Suppose that a primitive $E$ exists on such a neighborhood $U$. For $u\in U$ and smooth real directions $v,w$, the mixed second derivatives of $(s,t)\mapsto E(u+sv+tw)$ at the origin agree. After division by $n$, this equality reads
B4B9D0AD937E
11.2.5
\begin{equation} \int _ Xw\mathrm{dd}^{\mathrm{c}}v\wedge \omega _ u^{n-1} =\int _ Xv\mathrm{dd}^{\mathrm{c}}w\wedge \omega _ u^{n-1}. \end{equation}
By lemma 11.2.2,
\[ 0=\mathrm{d}(\omega _ u^{n-1}) =(n-1)\mathrm{d}\omega \wedge (\omega +\mathrm{dd}^{\mathrm{c}}u)^{n-2}. \]
Fix a point $x$. A real quadratic polynomial in local coordinates, multiplied by a cutoff function equal to one near $x$, realizes any prescribed real $(1,1)$-form $H$ as $\mathrm{dd}^{\mathrm{c}}u(x)$. These functions can be chosen to depend linearly on $H$, with global $C^2$ norms bounded by a fixed constant times the coefficient norm of $H$. Thus every sufficiently small $H$ is realized by a potential in $U$, and the preceding identity holds at $x$ with $\omega (x)+H$ ranging over an open neighborhood of the positive form $\omega (x)$. Applying lemma 11.2.3 with $T=\mathrm{d}\omega (x)$, we conclude that $\mathrm{d}\omega (x)=0$.
Since $x$ was arbitrary, $\mathrm{d}\omega =0$. Conversely, proposition 11.2.8 below constructs a primitive for a closed background. $\square$
In dimension one every real $(1,1)$-form is closed, and proposition 11.2.8 always gives a primitive. For $n\geq 2$, the symmetry condition at a single potential $u$ is $\mathrm d(\omega _ u^{n-1})=0$; the theorem uses it on an open set of potentials. Mass invariance alone does not suffice. Indeed, let $X$ be a compact complex surface admitting no Kähler metric, and let $\omega $ be a Gauduchon form on $X$, that is, $\mathrm{dd}^{\mathrm{c}}\omega =0$ (definition 11.4.1); such forms exist by theorem 11.4.2. The mass of $\omega $ is constant by corollary 11.1.10, but $\omega $ is not closed, so its Monge–Ampère one-form has no primitive.
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