Pome

Theorem 17.16.19B0CA576B1836

Theorem 17.16.19.B0CA576B1836 The completion of $(\mathcal B_\infty ,d_ B)$ is $\mathcal B_2$. It is a CAT$(0)$ space, and its unique constant-speed geodesic between two points is their full Hermitian Perron segment. This segment attains both endpoints in $L^1(X,\mathrm dV)$ for every smooth positive volume form $\mathrm dV$. For every $u\in \mathcal B_2$ and $c\in \mathbb R$,

\[ d_ B(u,u+c)=\sqrt{2n}|c|. \]

Proof. Fiber integration gives the exact restricted metric

D40B24211305 17.16.20
\begin{equation} G_{\pi ^*v}(\pi ^*\xi ,\pi ^*\eta ) =2n\int _ Y\xi \eta (\kappa +\mathrm{dd}^{\mathrm{c}}v)^ d. \end{equation}

The Kähler completion theorem therefore gives the stated completion and CAT$(0)$ property after multiplying base distances by $\sqrt{2n}$. If $\widetilde d_ D$ denotes the completed metric written with operator $\mathrm i\partial \bar\partial $ and background $\kappa $, the normalization is explicitly

58F89D956FBF 17.16.21
\begin{equation} d_ B(\pi ^*v,\pi ^*w) =2\pi \sqrt{2n}\, \widetilde d_ D(v/(2\pi ),w/(2\pi )), \end{equation}

because $\kappa +\mathrm i\partial \bar\partial (v/(2\pi ))=\kappa +\mathrm{dd}^{\mathrm{c}}v$. The preceding proposition identifies its unique geodesic with the full Hermitian Perron envelope. The base capacity endpoint limits give $L^1(Y)$ convergence. Since

\[ \int _ X|\pi ^*(v-w)|\frac{\omega ^ n}{2n} =\int _ Y|v-w|\kappa ^ d, \]

the same holds on $X$; every smooth positive $\mathrm dV$ has bounded density ratio with $\omega ^ n$ on compact $X$. For smooth $v$, the constant translation is a base geodesic with speed $|c|$. Smooth metric approximation and translation by the same constant extend the distance formula to every $v\in \mathcal E^2$. $\square$

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