Appendix A The skeleton and the retraction in the toric case by José Ignacio Burgos Gil and Martín Sombra [03AA]

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Appendix A The skeleton and the retraction in the toric case
by José Ignacio Burgos Gil and Martín Sombra

In this appendix we give a combinatorial description of the skeleton associated to a toric model of a toric variety, and of the corresponding retraction. We will use this description to show an example of two models of the same variety that have the same skeleton but different retractions. In turn this will give a counterexample to a higher dimensional extension of Proposition 3.8.

Let $(K,{|\phantom{a}|})$ be a complete non-archimedean discretely valued field, $K^{\circ}$ the valuation ring, $k$ the residue field, and $S=\Spec(K^{\circ})$ . Let $\varpi$ be a uniformizer of $K^{\circ}$ and write

\lambda_{K}=-\log|\varpi|.

Let $X$ be a smooth projective variety over $K$ of dimension $n$ and $X^{{\mathrm{an}}}$ the associated Berkovich analytic space. Let $\mathscr{X}$ be an SNC model of $X$ over $S$ , that is an SNC projective scheme ${{\mathscr{X}}}$ over $S$ with generic fiber $X$ such that the special fiber, which is not assumed to be reduced, agrees as a closed subset with a simple normal crossing divisor $D$ of ${{\mathscr{X}}}$ . To the model $\mathscr{X}$ we can associate an skeleton $\Delta_{\mathscr{X}}\subset{X^{{\mathrm{an}}}}$ and a retraction $p_{\mathscr{X}}\colon X^{{\mathrm{an}}}\to\Delta_{\mathscr{X}}$ , see [BFJ16a, §3] for details.

Let $L$ be an ample line bundle on $X$ and $\mathscr{L}$ a nef model of $L$ on $\mathscr{X}$ . Let $\theta$ be the semipositive $(1,1)$ -form in the class of $L$ corresponding to the model $\mathscr{L}$ . Let $\mu$ be a positive Radon measure on $X^{{\mathrm{an}}}$ with support in $\Delta_{\mathscr{X}}$ such that $\mu(X^{{\mathrm{an}}})=\deg_{L}(X)$ . The Monge-Ampère equation looks for a $\theta$ -psh function $\varphi$ on $X^{{\mathrm{an}}}$ such that

(A.1)

(dd^{c}\varphi+\theta)^{\wedge n}=\mu.

With the generality we are discussing in this paragraph, there is not yet a definition of the class of $\theta$ -psh functions with all the properties of classical pluripotential theory, but every good definition of this class should include the class of $\theta$ -psh model functions as introduced in 2.5.

The following question is natural and in case of being true would be of great help to solve the Monge-Ampère equation in positive and mixed charateristic.

Question 1.

With the previous hypothesis, is it true that any solution $\varphi$ to the Monge-Ampère equation (A.1) satisfies

(A.2)

\varphi=\varphi\circ p_{\mathscr{X}}?

We will see that this question has a negative answer by exhibiting a counterexample in the context of toric varieties. In fact, that this question has a negative answer is related with Proposition 3.8 not being true in higher dimension. To this aim, we will consider a smooth projective variety $X$ of dimension 2 and two models $\mathscr{X}$ and $\mathscr{X}^{\prime}$ that have the same skeleton

\Delta=\Delta_{\mathscr{X}}=\Delta_{\mathscr{X}^{\prime}}

but with different retractions $p_{\mathscr{X}}\not=p_{\mathscr{X}^{\prime}}$ . We will fix a semipositive $(1,1)$ -form $\theta$ that is realized in both models and construct two model functions $\varphi$ and $\varphi^{\prime}$ on $X^{{\mathrm{an}}}$ satisfying

(A.3)		$\displaystyle\varphi$	$\displaystyle\not=\varphi^{\prime},$	$\displaystyle\varphi\mid_{\Delta}$	$\displaystyle=\varphi^{\prime}\mid_{\Delta},$
(A.4)		$\displaystyle\varphi$	$\displaystyle=\varphi\circ p_{\mathscr{X}},$	$\displaystyle\varphi^{\prime}$	$\displaystyle=\varphi^{\prime}\circ p_{\mathscr{X}^{\prime}}.$

As a consequence of these properties, we deduce that $\varphi=\varphi^{\prime}\circ p_{\mathscr{X}}$ and that $\varphi^{\prime}=\varphi\circ p_{\mathscr{X}^{\prime}}$ . Moreover $\varphi^{\prime}$ will be a $\theta$ -psh model function while the model function $\varphi$ will not be $\theta$ -psh. Let $\mu\coloneqq(dd^{c}\varphi^{\prime}+\theta)^{\wedge 2}$ . This is a positive measure with support on $\Delta$ and $\varphi^{\prime}$ is a solution of the corresponding Monge-Ampère equation. Since

\varphi^{\prime}\not=\varphi=\varphi^{\prime}\circ p_{\mathscr{X}},

we see that $\varphi^{\prime}$ is a counterexample to Question 1 for the model $\mathscr{X}$ . Moreover, if Proposition 3.8 were true in dimension 2, then $\varphi=\varphi^{\prime}\circ p_{\mathscr{X}}$ would be a $\theta$ -psh model function, but it is not.

We place ourselves in the framework and notation of [BPS14]. The results below will make explicit the skeleton and the retraction associated to a toric SNC model of a toric variety.

Let $\mathbb{T}\simeq\mathbb{G}_{\rm m}^{n}$ be a split torus over $K$ . We denote by

M=\Hom(\mathbb{T},\mathbb{G}_{\rm m}),\quad N=\Hom(\mathbb{G}_{\rm m},\mathbb{T}),

the lattices of characters and one-parameter subgroups of $\mathbb{T}$ . Then $M=N^{\vee}$ . We also denote $N_{\mathbb{R}}=N\otimes\mathbb{R}$ and $M_{\mathbb{R}}=M\otimes\mathbb{R}$ . The pairing between $u\in N_{\mathbb{R}}$ and $x\in M_{\mathbb{R}}$ is denoted by $\langle x,u\rangle$ .

Let now $X$ be a proper toric variety over $K$ and $\mathscr{X}$ a proper toric model of $X$ over $S$ . Then $X$ is described by a complete fan $\Sigma$ in $N_{\mathbb{R}}$ and $\mathscr{X}$ is described by a complete SCR-polyhedral complex $\Pi$ whose recession fan satisfies $\rec(\Pi)=\Sigma$ [BPS14, Thm. 3.5.4].

There is a map $\zeta_{K}\colon N_{\mathbb{R}}\to\mathbb{T}^{{\mathrm{an}}}$ that sends $u\in N_{\mathbb{R}}$ to the seminorm on $K[M]$ given by

\left|\sum_{m\in M}\alpha_{m}\chi^{m}\right|=\max_{m}|\alpha_{m}|e^{-\lambda_{K}\langle m,u\rangle}.

This is a particular case of the map denoted by $\theta_{\sigma}$ in [BPS14, Prop.-Def. 4.2.12] composed with the homothety of ratio $\lambda_{K}$ .

There is also a map $\Val_{K}\colon\mathbb{T}^{{\mathrm{an}}}\to N_{\mathbb{R}}$ that sends a point $p\in\mathbb{T}^{{\mathrm{an}}}$ to the point $\Val_{K}(p)\in N_{\mathbb{R}}$ determined by

\langle m,\Val_{K}(p)\rangle=-\frac{1}{\lambda_{K}}\log|\chi^{m}(p)|,

see [BPS14, Sec. 4.1]. From the definition, it follows that $\Val_{K}\circ\,\zeta_{K}=\Id_{N_{\mathbb{R}}}$ .

To each polyhedron $\Lambda\in\Pi$ there is associated an orbit $O(\Lambda)$ for the action of $\mathbb{T}_{k}$ on the special fiber $\mathscr{X}_{s}$ [BPS14, §3.5]. We denote by $\xi_{\Lambda}$ the generic point of $O(\Lambda)$ .

The relation of $\Val_{K}$ with the reduction map is given by [BPS14, Cor. 4.5.2]: a point $p\in\mathbb{T}^{{\mathrm{an}}}$ satisfies ${\mathrm{red}}(p)\in O(\Lambda)$ if and only in $\Val_{K}(p)\in\rint(\Lambda)$ . The relation of $\zeta_{K}$ with the reduction map is given by the next result.

Lemma A.1.

Let $\Lambda\in\Pi$ . If $u$ lies in the relative interior of $\Lambda$ , then ${\mathrm{red}}(\zeta_{K}(u))=\xi_{\Lambda}$ .

Proof.

We use the notation of [BPS14, §3.5]. Let $\mathscr{X}_{\Lambda}$ be the affine toric scheme associated to $\Lambda$ . The ring of functions of $\mathscr{X}_{\Lambda}$ is

K^{\circ}[\mathscr{X}_{\Lambda}]=K^{\circ}[\widetilde{M}_{\Lambda}]/(\chi^{(0,1)}-\varpi).

The orbit $O(\Lambda)$ is a closed subscheme of $\mathscr{X}_{\Lambda}$ . If $u\in\rint(\Lambda)$ , the ideal of $O(\Lambda)$ is the ideal generated by the monomials $\chi^{(m,l)}$ with $(m,l)\in\widetilde{M}_{\Lambda}$ and $\langle m,u\rangle+l>0$ .

The generic fiber of $\mathscr{X}_{\Lambda}$ is the affine toric variety $X_{\rec(\Lambda)}=\Spec(K[M_{\rec(\Lambda)}])$ . The natural inclusion $K^{\circ}[\mathscr{X}_{\Lambda}]\subset K[M_{\rec(\Lambda)}]$ is given by $\chi^{(m,l)}\mapsto\varpi^{l}\chi^{m}$ . Any point $p\in X_{\rec(\Lambda)}^{{\mathrm{an}}}$ determines a seminorm on $K^{\circ}[\mathscr{X}_{\Lambda}]$ . The set of points of $X^{{\mathrm{an}}}_{\rec(\Lambda)}$ whose reduction belongs to $\mathscr{X}_{\Lambda}$ is

C=\{p\in X^{{\mathrm{an}}}_{\rec(\Lambda)}\mid|f(p)|\leq 1,\ \forall f\in K^{\circ}[\mathscr{X}_{\Lambda}]\}.

Given a point $p\in C$ , then ${\mathrm{red}}(p)$ is the point corresponding to the prime ideal

\mathfrak{q}_{p}=\{f\in K^{\circ}[\mathscr{X}_{\Lambda}]\mid|f(p)|<1\}.

Every $f\in K^{\circ}[\mathscr{X}_{\Lambda}]$ can be written as a sum

f=\sum_{(m,l)\in\widetilde{M}_{\Lambda}}\alpha_{(m,l)}\chi^{(m,l)},

with $|\alpha_{(m,l)}|=0,1$ and only a finite number of coefficients $\alpha_{(m,l)}$ different from zero.

By the definition of $\zeta_{K}$ ,

	$\displaystyle\mathfrak{q}_{\zeta_{K}(u)}$	$\displaystyle=\left\{\sum\alpha_{(m,l)}\chi^{(m,l)}\in K^{\circ}[\mathscr{X}_{\Lambda}]\,\middle\|\,\|\varpi\|^{l}e^{-\lambda_{K}\langle m,u\rangle}<1\right\}$
		$\displaystyle=\left\{\sum\alpha_{(m,l)}\chi^{(m,l)}\in K^{\circ}[\mathscr{X}_{\Lambda}]\,\middle\|\,\langle m,u\rangle+l>0\right\}.$

Since $u\in\rint(\Lambda)$ , we deduce that $\mathfrak{q}_{\zeta_{K}(u)}$ is the ideal of $O(\Lambda)$ and therefore ${\mathrm{red}}(\zeta_{K}(u))=\xi_{\Lambda}.$ ∎

We add now to $X$ the condition of being regular, which is equivalent to $\Sigma$ being unimodular, and to $\mathscr{X}$ the condition of being an SNC model. By [KKMS73, Chap. IV, §3.I item d)], $\mathscr{X}$ is regular if and only if the rational fan in $N_{\mathbb{R}}\times\mathbb{R}_{\geq 0}$ generated by $\Pi\times\{1\}$ is unimodular. In this case, the model is always an SNC model. On the other hand, by [BMPS16, Example 3.6.11] the model will be strictly semistable (SNC with reduced special fiber) if, in addition, all the vertices are lattice points.

Since a unimodular fan is necessarily simplicial, for each polyhedron $\Lambda\in\Pi$ , of dimension $t$ , we can write

(A.5)

\Lambda=\conv(p_{0},\dots,p_{s})+\cone(v_{s+1},\dots,v_{t}),

where $s\leq t$ , $p_{i}$ are points of $N_{\mathbb{R}}$ and $v_{i}$ are vectors on the tangent space to $N_{\mathbb{R}}$ at a point, that we identify with $N_{\mathbb{R}}$ .

We define the combinatorial skeleton as

\Delta_{\Pi}=\bigcup_{\begin{subarray}{c}\Lambda\in\Pi\\ \Lambda\text{ bounded}\end{subarray}}\Lambda.

There is a combinatorial retraction $p_{\Pi}\colon N_{\mathbb{R}}\to\Delta_{\Pi}$ defined as follows. Let $u\in N_{\mathbb{R}}$ and let $\Lambda\in\Pi$ be a polyhedron of dimension $t$ with $u\in\Lambda$ . Write $\Lambda$ as in equation (A.5). Therefore $u$ can be written uniquely as

(A.6)

u=\sum_{i=0}^{s}a_{i}p_{i}+\sum_{j=s+1}^{t}\lambda_{j}v_{j},

with $a_{i},\lambda_{j}\geq 0$ , and $\sum a_{i}=1$ . Then

p_{\Pi}(u)=\sum_{i=0}^{s}a_{i}p_{i}.

Remark A.2.

The fan $\Sigma$ determines a compactification $N_{\Sigma}$ of $N_{\mathbb{R}}$ as in [BPS14, §4.1] such that the retraction $p_{\Pi}$ can be extended to a continuous map $N_{\Sigma}\to\Delta_{\Pi}$ .

The following result explicites the skeleton and retraction associated to the model $\mathscr{X}$ .

Theorem A.3.

With the previous hypothesis, the skeleton $\Delta_{\mathscr{X}}\subset X^{{\mathrm{an}}}$ is given by

\Delta_{\mathscr{X}}=\zeta_{K}(\Delta_{\Pi}).

The restriction to $\mathbb{T}^{{\mathrm{an}}}$ of the retraction $p_{\mathscr{X}}$ is the composition

p_{\mathscr{X}}\mid_{\mathbb{T}^{{\mathrm{an}}}}=\zeta_{K}\circ p_{\Pi}\circ\Val_{K}.

Proof.

We start by recalling the construction of $\Delta_{\mathscr{X}}$ and $p_{\mathscr{X}}$ from [BFJ16a]. Note that, in loc. cit. the residue field $k$ is of characteristic zero, but once we assume that the model $\mathscr{X}$ is an SNC model, using the results of [MN15, § 3.1] it is possible to extend the presentation of [BFJ16a] to the case of positive and mixed characteristic.

Let $\Div_{0}(\mathscr{X})$ be the group of vertical Cartier divisors on $\mathscr{X}$ . Denote $\Div_{0}(\mathscr{X})_{\mathbb{R}}=\Div_{0}(\mathscr{X})\otimes\mathbb{R}$ and let $\Div_{0}(\mathscr{X})_{\mathbb{R}}^{\ast}$ be the dual. As explained in 2.2, each $D\in\Div_{0}(\mathscr{X})$ determines a model function $\varphi_{D}$ . The map $D\mapsto\varphi_{D}$ is linear in $D$ and can be extended by linearity to a map $\Div_{0}(\mathscr{X})_{\mathbb{R}}\to C^{0}(X^{{\mathrm{an}}})$ .

There is a map $\ev_{\mathscr{X}}\colon{X^{{\mathrm{an}}}}\to\Div_{0}(\mathscr{X})_{\mathbb{R}}^{\ast}$ determined by

(A.7)

\langle D,\ev_{\mathscr{X}}(x)\rangle=\varphi_{D}(x).

Let $D_{1},\dots,D_{\ell}$ be the components of the special fiber $\mathscr{X}_{s}$ . Each $D_{i}$ , $i=1,\dots,\ell$ , determines a divisorial point $x_{i}\in X^{{\mathrm{an}}}$ and we denote by $e_{i}=\ev_{\mathscr{X}}(x_{i})$ . For each $J\subset\{1,\dots,\ell\}$ we write $D_{J}=\bigcap_{j\in J}D_{j}$ and $\sigma_{J}=\conv(e_{j},j\in J)$ . Then the abstract skeleton of $\mathscr{X}$ is

\Delta^{\abs}_{\mathscr{X}}=\bigcup_{\begin{subarray}{c}J\subset\{1,\dots,\ell\}\\ D_{J}\not=\emptyset\end{subarray}}\sigma_{J}\subset\Div_{0}(\mathscr{X})_{\mathbb{R}}^{\ast}.

By [BFJ16a, Thm. 3.1], the image of $\ev_{\mathscr{X}}$ is $\Delta^{\abs}_{\mathscr{X}}$ and there exists a unique function $\emb_{\mathscr{X}}\colon\Delta^{\abs}_{\mathscr{X}}\to X^{{\mathrm{an}}}$ such that

(i)

$\ev_{\mathscr{X}}\circ\emb_{\mathscr{X}}=\Id_{\Delta^{\abs}_{\mathscr{X}}}$ ;
(ii)

for each $s\in\Delta_{\mathscr{X}}^{\abs}$ , if $s\in\rint(\sigma_{J})$ , then ${\mathrm{red}}(\emb_{\mathscr{X}}(s))=\xi_{D_{J}}$ , where $\xi_{D_{J}}$ is the generic point of $D_{J}$ .

Then the skeleton and the retraction are given by

\Delta_{\mathscr{X}}=\emb_{\mathscr{X}}(\Delta^{\abs}_{\mathscr{X}})\quad\quad p_{\mathscr{X}}=\emb_{\mathscr{X}}\circ\ev_{\mathscr{X}}.

We now go back to the regular toric case. In particular, $X$ is a toric smooth projective variety over $K$ and $\mathscr{X}$ is a toric projective SNC model. Then all the divisors of $\Div_{0}(\mathscr{X})$ are toric divisors. Therefore, for $D\in\Div_{0}(\mathscr{X})_{\mathbb{R}}$ , the function $\varphi_{D}$ is invariant under the action of the compact torus $\mathbb{S}=\Val_{K}^{-1}(0)$ . The restriction of $\varphi_{D}$ to $\mathbb{T}^{{\mathrm{an}}}$ factorizes as

(A.8)

\varphi_{D}\mid_{\mathbb{T}^{{\mathrm{an}}}}=-\phi_{D}\circ\Val_{K},

where $\phi_{D}$ is the function from [BPS14, Def. 4.3.6] corresponding to the trivial line bundle $\mathcal{O}_{X}$ with the metric determined by $D$ and the section $1$ .

We now define $\ev_{\Pi}\colon N_{\mathbb{R}}\to\Div_{0}(\mathscr{X})^{\ast}$ by

\langle D,\ev_{\Pi}(u)\rangle=-\phi_{D}(u).

By construction, the restriction of $\ev_{\Pi}$ to each polyhedron $\Lambda\in\Pi$ is affine. Moreover, using (A.7) and (A.8) we deduce that

(A.9)

\ev_{\mathscr{X}}\mid_{\mathbb{T}^{{\mathrm{an}}}}=\ev_{\Pi}\circ\Val_{K}.

As before let $D_{1},\dots,D_{\ell}$ be the components of the special fiber $\mathscr{X}_{s}$ and $x_{i}$ the divisorial point determined by $D_{i}$ . Then the set of vertices of $\Pi$ is $\Pi^{0}=\{u_{1},\dots,u_{\ell}\}$ , where $u_{i}=\Val_{K}(x_{i})$ . Therefore $\ev_{\Pi}(u_{i})=e_{i}$ . Since $\ev_{\Pi}$ is affine in each polyhedron of $\Pi$ we deduce that the image of $\ev_{\Pi}$ is $\Delta^{\abs}_{\mathscr{X}}$ and that $\ev_{\Pi}$ determines a homeomorphism $\Delta_{\Pi}\to\Delta^{\abs}_{\mathscr{X}}$ . We define $\emb_{\Pi}\colon\Delta^{\abs}_{\Pi}\to N_{\mathbb{R}}$ as the composition of the inverse of this homeomorphism with the inclusion $\Delta_{\Pi}\hookrightarrow N_{\mathbb{R}}$ . Using equation (A.9) and Lemma A.1 one can check that $\zeta_{K}\circ\emb_{\Pi}$ satisfies the conditions (1) and (2) that characterize $\emb_{\mathscr{X}}$ . Therefore

(A.10)

\emb_{\mathscr{X}}=\zeta_{K}\circ\emb_{\Pi}.

We next claim that $p_{\Pi}=\emb_{\Pi}\circ\ev_{\Pi}.$ Indeed, for every $D\in\Div_{0}(\mathscr{X})$ , since $D$ is a model of the trivial vector bundle, we know that $\rec(\phi_{D})$ is the zero function. Therefore, writing any $u\in\Lambda\in\Pi$ is as in (A.6), one can show that

\phi_{D}=\phi_{D}\circ p_{\Pi}.

This implies that $\ev_{\Pi}=\ev_{\Pi}\circ\,p_{\Pi}$ . By construction $\emb_{\Pi}\circ\ev_{\Pi}$ is the identity in the image of $p_{\Pi}$ . Therefore

(A.11)

\emb_{\Pi}\circ\ev_{\Pi}=\emb_{\Pi}\circ\ev_{\Pi}\circ p_{\Pi}=p_{\Pi}.

Using equations (A.11) (A.10) and (A.9) we deduce that

\Delta_{\mathscr{X}}=\emb_{\mathscr{X}}(\Delta^{\abs}_{\mathscr{X}})=\zeta_{K}(\emb_{\Pi}(\Delta^{\abs}_{\mathscr{X}}))=\zeta_{K}(\Delta_{\Pi})

and

p_{\mathscr{X}}|_{\mathbb{T}^{{\mathrm{an}}}}=\emb_{\mathscr{X}}\circ\ev_{\mathscr{X}}|_{\mathbb{T}^{{\mathrm{an}}}}=\zeta_{K}\circ\emb_{\Pi}\circ\ev_{\Pi}\circ\Val_{K}=\zeta_{K}\circ p_{\Pi}\circ\Val_{K}

concluding the proof. ∎

Figure 1. Subdivisions corresponding to the toric models

\mathscr{X}

and

\mathscr{X}^{\prime}

Consider the toric variety $X=\mathbb{P}_{K}^{2}$ . Let $\mathbb{P}^{2}_{S}$ be the projective space over $S$ , and let $(x_{0}:x_{1}:x_{2})$ be homogeneous coordinates of the special fiber $\mathbb{P}^{2}_{k}$ . Consider the model $\mathscr{X}$ of $X$ obtained by blowing up $\mathbb{P}^{2}_{S}$ at the line $x_{2}=0$ inside the special fiber and then blowing up the strict transform of the line $x_{1}=0$ . The SCR-polyhedral subdivision $\Pi$ associated to this model is depicted in the left side of figure 1. Consider also the model $\mathscr{X}^{\prime}$ of $X$ obtained as before, switching $x_{1}$ and $x_{2}$ . The SCR-polyhedral subdivision $\Pi^{\prime}$ associated to this new model is depicted in the right side of figure 1. Both toric schemes $\mathscr{X}$ and $\mathscr{X}^{\prime}$ are SNC models (even more, they are strictly semistable models) of $\mathbb{P}^{2}_{K}$ .

The skeleton associated to both models is the simplex

\Delta=\conv((0,0),(1,0),(0,1))

and both retractions $p_{\Pi}$ and $p_{\Pi^{\prime}}$ are also depicted on the same figure. For instance the retraction $p_{\Pi}$ sends every point of $\sigma_{2}$ to the point $(1,0)$ , while the same retraction restricted to the polyhedron $\sigma_{1}$ is the horizontal projection onto the segment $\overline{(0,1)(1,0)}$ along the direction $(-1,0)$ . By contrast, the retraction $p_{\Pi^{\prime}}$ sends both cones $\sigma_{1}^{\prime}$ and $\sigma_{2}^{\prime}$ to the point $(1,0)$ .

Consider the divisor $D$ of $\mathbb{P}_{K}^{2}$ given by the line at infinity and the divisor $\mathcal{D}$ of $\mathbb{P}_{S}^{2}$ given by the closure of $D$ . Let $L=\mathcal{O}_{\mathbb{P}_{K}^{2}}(D)$ and $\mathscr{L}=\mathcal{O}_{\mathbb{P}^{2}_{S}}(\mathcal{D})$ . Then $\mathscr{L}$ is a model of $L$ in $\mathbb{P}_{S}^{2}$ and can be pulled back to both $\mathscr{X}$ and $\mathscr{X}^{\prime}$ . Let $\theta$ be the closed $(1,1)$ -form defined by this model.

Let $\Psi\colon N_{\mathbb{R}}\to\mathbb{R}$ be the function

\Psi(u,v)=\min(u,v,0).

This is the function that determines the toric divisor $D$ .

By [BPS14, Thm. 4.8.1], the space of all continuous $\theta$ -psh functions on $X^{{\mathrm{an}}}$ that are invariant under the action of the compact torus $\mathbb{S}$ can be identified with the set of all bounded functions $f\colon N_{\mathbb{R}}\to\mathbb{R}$ such that $\Psi+f$ is concave. This identification sends $f\colon N_{\mathbb{R}}\to\mathbb{R}$ to the unique continuous function $\varphi\colon X^{{\mathrm{an}}}\to\mathbb{R}$ such that $\varphi\mid_{\mathbb{T}^{{\mathrm{an}}}}=-f\circ\Val_{K}$ .

Let $g\colon\Delta\to\mathbb{R}$ the affine function that has the value $1$ at the point $(1,0)$ and the value $0$ at the points $(0,0)$ and $(0,1)$ and put

f=g\circ p_{\Pi},\qquad f^{\prime}=g\circ p_{\Pi^{\prime}}.

One easily verifies that

\Psi+f^{\prime}=\min(1,1+v,u)

which is concave. On the other hand, the restriction of $\Psi+f$ to $\sigma_{3}$ is $0$ while its restriction to $\sigma_{1}$ is $1-v$ , hence $\Psi+f$ is not concave.

Let $\varphi^{\prime}$ be the continuous function on $X^{{\mathrm{an}}}$ whose restriction to $\mathbb{T}^{{\mathrm{an}}}$ is $-f^{\prime}\circ\Val_{K}$ . It is a model $\theta$ -psh function. The function $-f\circ\Val_{K}$ also extends to a model function $\varphi$ on $X^{{\mathrm{an}}}$ but it is not $\theta$ -psh because $f$ is not concave [BPS14, Thm. 3.7.1 (2)].

We now write

\mu=(dd^{c}\varphi^{\prime}+\theta)^{\wedge 2}.

By [BPS14, Thm. 4.7.4] the measure $\mu$ is the atomic measure with support in $\zeta_{K}((1,0))$ with total mass one. Hence its support is contained in $\Delta=\Delta_{\mathscr{X}}$ .

Summing up, $\mu$ is a measure with support in $\Delta=\Delta_{\mathscr{X}}$ , the $\theta$ -psh function $\varphi^{\prime}$ is a solution of the corresponding Monge-Ampère equation but $\varphi^{\prime}\not=\varphi^{\prime}\circ p_{\mathscr{X}}$ showing that the answer to Question 1 is negative. Moreover $\varphi=\varphi^{\prime}\circ p_{\mathscr{X}}$ is not $\theta$ -psh, showing that Proposition 3.8 does not extend to dimension $\geq 2$ .

J.I. Burgos Gil, Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UCM3), Calle Nicolás Cabrera 15, Campus de la Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
E-mail address: burgos@icmat.es

M. Sombra, Institució Catalana de Recerca i Estudis Avançats (ICREA). Passeig Lluís Companys 23, 08010 Barcelona, Spain
Departament de Matemàtiques i Informàtica, Universitat de Barcelona (UB). Gran Via 585, 08007 Barcelona, Spain
E-mail address: sombra@ub.edu