Theorem 14.26.1.050594389138 On $X=\mathbb P^3$ there exist a smooth Hermitian form $\omega $ with $\mathrm{d}\omega \ne 0$, a smooth nonnegative nonzero function $f$, and distinct smooth functions $u,v\in \mathrm{PSH}(X,\omega )$ such that
\[ \sup _ Xu=\sup _ Xv=0,\qquad (\omega +\mathrm{dd}^{\mathrm{c}}u)^3=f\omega _ X^3=(\omega +\mathrm{dd}^{\mathrm{c}}v)^3. \]
The density vanishes on a nonempty open set, and both potentials have minimal singularities.
of theorem 14.26.1.
Set $\widetilde u=-w/2-\rho $ and $\widetilde v=w/2-\rho $. Since $\mathrm{dd}^{\mathrm{c}}w=2H$, the construction gives
\[ \omega +\mathrm{dd}^{\mathrm{c}}\widetilde u=A_0\geq 0,\qquad \omega +\mathrm{dd}^{\mathrm{c}}\widetilde v=B_0\geq 0. \]
Subtracting their respective suprema gives $u,v$. Their Hessians are unchanged, and $v-u=w+\text{constant}$ is nonconstant. The function $f=A_0^3/\omega _ X^3=B_0^3/\omega _ X^3$ is smooth, nonnegative and nonzero, and it vanishes on the nonempty open band $|r|<1$. Both potentials are smooth, hence bounded, so they have minimal singularities: for a Hermitian background, the envelope $V_\omega $ of section 5.40 is zero.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.