Example 12.10.1536B0532E29A1

Proof. Fix a Gauduchon metric $\gamma $ and a point $x$. Choose a smooth real function $\psi $ such that $\mathrm{dd}^{\mathrm{c}}\psi \leq -\gamma $ on a coordinate neighborhood $U$ of $x$; in holomorphic coordinates $z=(z_1,\ldots ,z_ n)$ centered at $x$, such a function is obtained by multiplying $-M|z|^2$, with $M$ large enough, by a cutoff that equals one near $x$, and then shrinking $U$.

Choose a smooth function $\chi $ with $0\leq \chi \leq 1$ that vanishes on a smaller neighborhood of $x$ and equals one outside $U$. Put

Since $X$ is compact, $\omega $ is strictly positive for sufficiently large $L>0$: on $U$ the term $-\mathrm{dd}^{\mathrm{c}}\psi $ is at least $\gamma $, and outside $U$ the term $L\gamma $ dominates $\mathrm{dd}^{\mathrm{c}}\psi $. Set $f=a^ n\gamma ^ n/\mathrm{d}V$. The smooth function $\psi $ satisfies $\omega +\mathrm{dd}^{\mathrm{c}}\psi =a\gamma \geq 0$ and $\mathrm{MA}_\omega (\psi )=f\, \mathrm{d}V$, so $b_\omega (f)\geq 1$.

Stokes' theorem and $\mathrm{dd}^{\mathrm{c}}\gamma ^{n-1}=0$ give

Testing (15.29.3) with $\gamma $ therefore gives the opposite inequality $b_\omega (f)\leq 1$. Hence $b_\omega (f)=1$, the metric $\gamma $ attains the infimum, and the normalized current $\gamma ^{n-1}/\int _ X\omega \wedge \gamma ^{n-1}$ is a smooth strictly positive maximizer of $D_{f^{1/n}}$. If $\omega $ were closed, corollary 12.10.14 would force $f$ to be strictly positive, contrary to its vanishing on an open set. Thus $\omega $ is nonclosed. $\square$