Proposition 17.10.650C8741A27906

Proof. By an affine change of parameter, we may assume that $[a,b]=[0,1]$.

Step 1. Assume first that $\varphi _0,\varphi _1\in \mathcal{E}^{\infty }(X,\theta ;\phi )$. Let $Y=X\times \mathbb {P}^1$, and let $p_1,p_2$ be the two projections. On

the function $U$ is $p_1^*\theta $-psh. Choose $C>0$ such that

The constant subgeodesic at $\phi -C$ and proposition 17.10.7 give

Fix concentric annuli $A_0\Subset A_1\Subset A$ and a unit-mass Fubini–Study form $\omega _{\mathrm{FS}}$ on $\mathbb {P}^1$. Put

The class represented by $\Theta _ B$ is big. Since $\phi =P_\theta [\phi ]$, we have $\phi \leq 0$. For every $B>0$, $\Phi =P_{\Theta _ B}[\Phi ]$. Indeed, if $\Psi \in \mathrm{PSH}(Y,\Theta _ B)$ satisfies $\Psi \leq 0$ and $\Psi \preceq \Phi $, then on each fibre $X\times \{ z\} $, either $\Psi _ z\equiv -\infty $, or $\Psi _ z\in \mathrm{PSH}(X,\theta )$ is a candidate in the definition of $P_\theta [\phi ]=\phi $. Thus $\Psi _ z\leq \phi $ in both cases. Conversely, $\Phi $ itself is a candidate. Moreover, proposition 13.11.4 6 gives

Choose an $S^1$-invariant smooth function $q$ on $\mathbb {P}^1$ such that

After increasing $B$, we may assume that $B\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q\geq 0$. Identifying $q$ with $p_2^*q$, define

The gluing lemma corollary 5.14.20 shows that $W\in \mathrm{PSH}(Y,\Theta _ B)$. By construction,

for some $C'>0$. Hence $W\in \mathcal{E}^{\infty }(Y,\Theta _ B;\Phi )$ by (17.10.67).

Set

Define the order-zero current on $\mathbb {P}^1$

The integration-by-parts formula [Xia19b], applied against a smooth test function on $\mathbb {P}^1$, gives

By (16.5.11), (17.10.68), and corollary 13.11.15, on $A_0$ the current $\mathcal{H}_ W$ is represented almost everywhere by

This representative is continuous. Indeed, on every smaller annulus, the convex functions $t\mapsto \varphi _ t(x)-\phi (x)$, considered on the common locus where both values are finite, are uniformly bounded by (17.10.66), hence uniformly Lipschitz. The resulting inequalities between the potentials hold almost everywhere and therefore everywhere by proposition 5.14.6. The monotonicity in 16.5.14 and (16.5.13) then give the local Lipschitz continuity of $t\mapsto E_\theta ^\phi (\varphi _ t)$.

On $X\times A_0$, define the signed Radon measure

This measure is positive. To see this, take a relatively compact product coordinate ball $Q\subset X\times A_0$. Let $a$ be a smooth local potential of $p_1^*\theta $, put $h=U+a\leq 0$ and $h_ k=h\lor (-k)$, and choose a smooth local potential $b$ of $B\beta $ with $|b|\leq C_ Q$. For $p=n,n+1$, set

Plurifine locality and $|b|\leq C_ Q$ give, for $k>C_ Q$,

Thus the two cofinal truncation families have the same limit. By proposition 13.11.4 6 and $\beta ^2=0$,

Passing to the increasing truncation limit yields

By (13.11.17),

since $W_ z\sim \phi $ on every fibre and proposition 16.3.10 applies. On the other hand,

by proposition 13.11.4 6. Together with (17.10.70) and (17.10.71), these identities give on $A_0$

Step 2. We prove that $\mu _ U=0$. Let $D\Subset X\times A$ be a coordinate domain, let $a$ be a smooth local potential of $p_1^*\theta $ near $\overline D$, and set $h=U+a$. If $v$ is a psh function on $D$ with the boundary inequality required in the definition of maximality, max-pasting gives

The function $\widetilde U$ need not be radial. Its $S^1$-symmetrization

is radial and $p_1^*\theta $-psh. The $S^1$-saturation of $D$ is relatively compact in $X\times A$, so $V=U$ near $X\times \partial A$. Hence the family $V_ t(x)=V(x,\mathrm{e}^{-t})$ is a subgeodesic from $\varphi _0$ to $\varphi _1$ by definition 17.10.2. The definition (17.10.21) therefore gives

It follows that $v\leq h$ on $D$, so $h$ is a maximal psh function.

On a product coordinate ball $Q\Subset X\times A_0$, put $O_ k=Q\cap \{ h>-k\} $. By [Proposition 5.9.3, EKF], $h$ is $\mathcal F$-maximal. The function $h_ k=h\lor (-k)$ is locally bounded and agrees with $h$ on the $\mathcal F$-open set $O_ k$, hence is $\mathcal F$-maximal there. Therefore [Theorem 5.10.1, EKF] gives

By (17.10.73), $\mu _ U=0$. Since $A_0\Subset A$ was arbitrary, (17.10.74) shows that $z\mapsto E_\theta ^\phi (\varphi _{-\log |z|})$ is radial and harmonic on $A$. Thus $t\mapsto E_\theta ^\phi (\varphi _ t)$ is affine on $(0,1)$.

Choose $C_0>0$ such that $|\varphi _1-\varphi _0|\leq C_0$ on their common finite locus; equivalently, the two resulting inequalities between the potentials hold everywhere. Affine barriers and proposition 17.10.7 give

By 16.5.14 and (16.5.13), the energy converges to the corresponding endpoint energy as $t\to 0+$ and $t\to 1-$. Hence

when the endpoints belong to $\mathcal E^\infty (X,\theta ;\phi )$.

Step 3. Let now $\varphi _0,\varphi _1\in \mathcal E^1(X,\theta ;\phi )$. By example 17.10.23 and (16.5.13), we may add different constants to the endpoints and assume that $\varphi _0,\varphi _1\leq \phi $. Put

and let $(\varphi _ t^{(k)})_{t\in [0,1]}$ be the corresponding geodesic. The endpoint truncations decrease to $\varphi _ j$, and Perron monotonicity gives

Let $h_ t=\lim _{k\to \infty }\varphi _ t^{(k)}$. The decreasing limit of the complexifications is $p_1^*\theta $-psh and is not identically $-\infty $. Taking the limit in the convex upper bound gives, first almost everywhere and then everywhere by proposition 5.14.6,

Note that

Since $\varphi _ t\to \varphi _0$ in $L^1$ as $t\to 0+$ and $\varphi _0,\varphi _1\in L^1(X)$, it follows that $h_ t\to \varphi _0$ in $L^1$. The same argument at $t=1$ gives $h_ t\to \varphi _1$ in $L^1$. Together with the $p_1^*\theta $-psh complexification constructed above, these endpoint limits show by definition 17.10.2 that $(h_ t)$ is a subgeodesic from $\varphi _0$ to $\varphi _1$. Hence (17.10.21) gives $h_ t\leq \varphi _ t$. Together with (17.10.76), this proves

Applying proposition 16.5.37 at the endpoints and at each interior slice, and then using (17.10.75), gives

This concludes the proof. $\square$