Let $I$ be the set of vertices in $\Delta$, let $L\subset I$ be the set of vertices contained in $\sta_{\Delta}(\sigma)$ and $J\subset L$ the set of vertices of $\sigma$. Thus $\sigma=\sigma_{J}$.

Consider the simplicial projective subdivision $\Delta^{\prime}=\Delta^{\prime}(\varepsilon)$ constructed in §6.2. For $j\in L$, $e^{\prime}_{j}:=\varepsilon e_{j}+(1-\varepsilon)v$ is a vertex of $\Delta^{\prime}$. Recall that $\sigma^{\prime}=\sigma^{\prime}_{J}$ is the face of $\Delta^{\prime}$ containing $v$ in its relative interior. Since $\varphi|_{\sigma}$ is assumed affine in a neighborhood of $v$, we may choose $\varepsilon>0$ small enough that:

• •

  $\varphi$ is affine on $\sigma^{\prime}\subset\sigma$

• •

  $\varphi$ is affine on each segment $[v,e^{\prime}_{j}]$, $j\in L$.

Let $\rho:\mathcal{X}^{\prime}\to\mathcal{X}$ be the vertical blow-up corresponding to the subdivision $\Delta^{\prime}$ of $\Delta$ as in Theorem 3.11. Note that $\rho$ induces a generically finite map $E^{\prime}_{J}\to E_{J}$ of projective $k$-varieties. Indeed, $E_{J}$ (resp. $E^{\prime}_{J}$) is the closure of the center of $v$ on $\mathcal{X}$ (resp. $\mathcal{X}^{\prime}$), and both have codimension $|J|$ by Theorem 3.11.

Recall that $\varphi=\varphi_{G}$ for some $G\in\Div_{0}(\mathcal{Y})$. We may assume that the determination $\mathcal{Y}$ of $\varphi$ dominates $\mathcal{X}^{\prime}$, so that $\pi$ factors as $\pi=\rho\circ\mu$ with $\mu:\mathcal{Y}\to\mathcal{X}^{\prime}$. As we shall see shortly, a first computation shows:

The key observation is now the following positivity property:

Grant this result for the moment. Lemma 6.4 and the projection formula yield

$$\deg(\rho|_{E^{\prime}_{J}})E_{J}\cdot\left(\theta_{\mathcal{X}}+\varphi(v)\,\mathcal{X}_{0}+\sum_{j\in L}D_{v}\varphi(e_{j})\,b_{j}E_{j}\right)\cdot\mathcal{A}^{n-p-1}=E^{\prime}_{J}\cdot(\rho^{*}\theta_{\mathcal{X}}+\mu_{*}G)\cdot\rho^{*}\mathcal{A}^{n-p-1}.$$

Here the right-hand side is non-negative by Lemma 6.5, since $\rho^{*}\mathcal{A}$ is nef, and we get

(6.3)

$$\sum_{j\in L}D_{v}\varphi(e_{j})\,b_{j}\left(E_{J}\cdot E_{j}\cdot\mathcal{A}^{n-p-1}\right)\geq-(E_{J}\cdot\theta_{\mathcal{X}}\cdot\mathcal{A}^{n-p-1})-\varphi(v)(E_{J}\cdot\mathcal{X}_{0}\cdot\mathcal{A}^{n-p-1}).$$

By induction, the $C^{0,1}$-norm of $\varphi|_{\sigma}$ is under control. Since $v$ belongs to $\sigma=\sigma_{J}$, this gives

$$|\varphi(v)|\leq C\text{ and }\max_{j\in J}|D_{v}\varphi(e_{j})|\leq C,$$

and (6.3) yields a lower bound

$$\sum_{j\in L\setminus J}D_{v}\varphi(e_{j})\,b_{j}\left(E_{J}\cdot E_{j}\cdot\mathcal{A}^{n-p-1}\right)\geq-C.$$

Now the convexity of $\varphi$ and the normalization $\sup_{X}\varphi=0$ show that

$$\max_{j\in L\setminus J}D_{v}\varphi(e_{j})\leq\max_{j\in L\setminus J}(\varphi(e_{j})-\varphi(v))\leq-\varphi(v)\leq C.$$

Here $E_{j}|_{E_{J}}$ is a non-zero effective divisor for $j\notin J$, hence $E_{J}\cdot E_{j}\cdot\mathcal{A}^{n-p-1}>0$ since $\mathcal{A}$ is ample. The previous inequality therefore implies, as desired, that $D_{v}\varphi(e)\geq-C$, since $e=e_{j}$ for some $j\in L\setminus J$. ∎