Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy

Definition A.1.

Let ϕ 0 , ϕ 1 be two continuous L-psh metrics. Their relative I-energy is defined as

Their relative J-energy is defined as

Given a reference metric ϕ ref , we write

The functionals I and J then, much like E, admit an extension to ℰ 1 ⁢ ( L ) , which is continuous along decreasing nets.

Proposition A.2.

Let ϕ 0 , ϕ 1 ∈ E 1 ⁢ ( L ) . If E ⁢ ( ϕ 0 , ϕ 1 ) = 0 and ϕ 0 ≥ ϕ 1 , then ϕ 0 = ϕ 1 .

Proof.

The main argument has been communicated to the author by S. Boucksom and M. Jonsson, as part of works on finite-energy spaces currently in writing.

Approximate ϕ 0 and ϕ 1 by decreasing nets ϕ 0 k , ϕ 1 k ∈ ℋ ⁢ ( L ) . Up to taking the maximum of the two sequences, we can assume without loss of generality that for all k, ϕ 0 k ≥ ϕ 1 k . We have

in the notations of Section 3.3. Since ϕ 0 ≥ ϕ 1 , all of the terms in the above sum are integrals against positive measures of nonnegative functions, hence they are all positive. In particular,

where the vanishing follows from continuity of E along decreasing nets, and the fact that E ⁢ ( ϕ 0 , ϕ 1 ) = 0 .

Pick any positive measure μ that can be expressed as MA ⁡ ( ϕ ) for some ϕ ∈ ℋ ⁢ ( L ) , and write for x ∈ X an ,

By the nontrivially-valued version of the estimate [16, Lemma 7.30] (which is proved similarly as in the trivially-valued case), given four Fubini–Study metrics ϕ i ∈ ℋ ⁢ ( L ) , i ∈ { 0 , 1 , 2 , 3 } , there exists constants C , a , b depending only on dim ⁡ X such that

In our case, we then have

recalling that we have defined μ = MA ⁡ ( ϕ ) . Now, by the continuity of the extensions of I and J along decreasing nets,

while we have established before that

We then find that

and the right-hand side vanishes, while the left-hand side converges as k → ∞ to the nonnegative quantity

The key point now is to find a measure μ = MA ⁡ ( ϕ ) with positive Dirac mass at x. Now, we recall that a psh function is uniquely determined by its restriction to the set of divisorial points in X an , and that for any such point x we may find a Monge–Ampère measure μ x associated to a projective model of X which has an atom at x, as in [9, Example 8.11]. As we then have

and μ x ⁢ ( { x } ) > 0 , we have that ϕ 0 = ϕ 1 on all divisorial points of X an , hence on X an . ∎