Pome

Example 13.22.34B14FA731306

Example 13.22.3.4B14FA731306 For every $n\geq 2$ there are a closed positive $(1,1)$-form $\theta $ on the projective manifold $\mathbb P^1\times \mathbb P^{n-1}$ and two ordinary models $a\not\sim b$ in $\mathrm{PSH}(\mathbb P^1\times \mathbb P^{n-1},\theta )$ such that

\[ P_\theta [a]_{\mathcal I}=b,\qquad \underline{\operatorname {vol}}_\theta (a) =\underline{\operatorname {vol}}_\theta (b)=0,\qquad d_\theta (a,b)=0. \]

For every $n\geq 3$, the same assertions hold for a suitable nonclosed positive form $\theta $ on a projective manifold of dimension $n$ with the bounded mass property.

Proof. Take the closed positive product background of example 18.15.4 on $\mathbb P^1\times \mathbb P^{n-1}$. Its models satisfy $a\leq b$ and $U(a)=U(b)=0$. Thus $\max (a,b)=b$, and the defining formula gives

\[ d_\theta (a,b)=2U(b)-U(a)-U(b)=0. \]

The other assertions are part of that example. For the nonclosed case use example 18.15.5; its models are again ordered and have zero upper volumes, so the same calculation applies. $\square$

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