Theorem 14.28.1.9C11A9E88B60 On $\mathbb P^3$ there exist a smooth nonclosed Hermitian form $\omega $ and a smooth nonnegative density $g$ such that:
The zero set of $g$ consists of two disjoint smooth real hypersurfaces. Near either hypersurface, $g$ is the square of a smooth defining function times a smooth strictly positive function.
The equation $(\omega +\mathrm{dd}^{\mathrm{c}}u)^3=g\omega _ X^3$ has uncountably many distinct smooth $\omega $-plurisubharmonic solutions normalized by $\sup _ Xu=0$.
The background $\omega $ has bounded mass.
of theorem 14.28.1.
Let $\eta _0>0$ be such that lemma 14.28.3 applies for $0<\eta <\eta _0$, and let $h_\eta $ denote the solution with $h_\eta (0)=-\eta $. Take a primitive $G_\eta $ of its extension by zero. Since $G_\eta $ is constant near each end of the real line, $G_\eta (r)$ defines a smooth function on $X$. The potential $u_\eta =G_\eta (r)-\rho $ satisfies
\[ \omega +\mathrm{dd}^{\mathrm{c}}u_\eta =(x+h_\eta )P+(y-h_\eta )Q+(z+h_\eta ')R. \]
Inside the band, the first two coefficients are positive by the lemma, and the equation gives
\[ z+h_\eta '=\frac{xyz}{(x+h_\eta )(y-h_\eta )}>0. \]
Outside the band the form is $\omega _0$, and on the two boundary hypersurfaces it is also the rank-two form $\omega _0$, since $h_\eta $ vanishes to infinite order there. Thus it is semipositive on $X$, and by (14.26.4) its cube is $6xyzP\wedge Q\wedge R=\omega _0^3=g\omega _ X^3$. Different initial values give potentials that do not differ by constants, so after normalization by the supremum we obtain an uncountable family. Bounded mass follows from proposition 13.14.15, as for theorem 14.26.1.
$\square$
Pome
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