Theorem 15.32.1D988D47D5E0E

Proof. For a Gauduchon form $\gamma $ on $X$, write $A_\gamma (f)=\int _ X(f\, \mathrm{d}V/\gamma ^ n)^{1/n}\gamma ^ n$. Since $|s^{1/n}-t^{1/n}|\leq |s-t|^{1/n}$ for $s,t\geq 0$, Hölder's inequality shows that $A_\gamma $ is continuous on $L^1$; it is positive at every nonzero nonnegative density. By theorem 15.29.1, $b_ h^{1/n}$ is the infimum over $\gamma $ of the continuous positive functions $f\mapsto \int _ Xh\wedge \gamma ^{n-1}/A_\gamma (f)$. It is therefore upper semicontinuous, and testing a single $\gamma $ bounds it above on an $L^1$ neighborhood of $f$.

For the second assertion, recall that $b_ h(q)=c_ h(q)$ for bounded $q$ (proposition 15.28.7). By the first assertion, $\limsup _ jc_ h(q_ j)\leq b_ h(f)$ for every sequence of nonzero nonnegative bounded densities $q_ j\to f$ in $L^1$. Hence the limit in (15.32.2) is at most $b_ h(f)$, and the local upper bound makes the suprema finite for small $\delta $. Conversely, the truncations $f_ k$ converge to $f$ in $L^1$ and satisfy $c_ h(f_ k)\geq b_ h(f)$, so every supremum in (15.32.2) is at least $b_ h(f)$. $\square$