Example 11.2.24.F9D5DB66D92A For $0<s<1$, let $\widehat\omega _ s$ on $X=S\times S$ be the sum of the pullbacks of $\alpha _ s$ from the first factor and of $\eta $ from the second, and set $V_0=\int _ S\mu _0$. Then $\mathrm d\widehat\omega _ s\ne 0$ and
\[ F(\widehat\omega _ s)=s^2V_0^2,\qquad V(\widehat\omega _ s)=(1-s^2)V_0^2. \]
The energy is constant on all positive smooth potential variations of $\widehat\omega _ s$, and every bounded $\widehat\omega _ s$-plurisubharmonic function has mass $24(1-s^2)V_0^2$. The form $\widehat\omega _ s$ is not a critical point of $F$ in its Aeppli class, and its Monge–Ampère one-form has no primitive on any $C^2$-open neighborhood of zero.
Proof.
The pullback of $\rho _ s$ from the first factor is a completion of $\widehat\omega _ s$. It is canonical: pure and mixed holomorphic two-forms are orthogonal for the product metric, and its pairing with pure-factor forms vanishes by the Stokes argument in the proof of example 11.2.23. The energy is therefore $\int _ X\rho _ s\wedge \bar\rho _ s\wedge \eta ^2/2=s^2V_0^2$. The volume is the product of the two surface volumes, which gives the second formula.
The exterior derivatives of $\widehat\omega _ s$ come entirely from the first factor. Their wedge product $\mathrm d\widehat\omega _ s\wedge \mathrm d^{\mathrm c}\widehat\omega _ s$ vanishes for degree reasons, while $\mathrm{dd}^{\mathrm{c}}\widehat\omega _ s=0$. Thus $\mathrm{dd}^{\mathrm{c}}(\widehat\omega _ s^2)=0$ by lemma 11.1.3, and corollary 11.2.21 gives the constancy of the energy and the common bounded mass.
The forms $\widehat\omega _{(1+t)s}$ are positive for small $|t|$ and lie in the Aeppli class of $\widehat\omega _ s$, since $v_{(1+t)s}=(1+t)v_ s$. The derivative of their energy at $t=0$ is $2s^2V_0^2>0$, so $\widehat\omega _ s$ is not a critical point in its Aeppli class. Finally, $\partial \alpha _ s=-\bar\partial \rho _ s\ne 0$. Hence $\mathrm d\widehat\omega _ s\ne 0$, and theorem 11.2.4 shows that no primitive exists.
$\square$
Pome
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