Subsection 3.5 (00TK) · ScalingStacks

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3.5 Extension property: the Fermat case

We do not know the equivalence between the extension property and the local convexity property. However, in the case of the Fermat family Example 3.1, the polyhedral set $\partial\Delta_{\lambda}^{\vee}=-\partial\Delta^{\vee}$ has a discrete symmetry by the permutation group of the vertices of $\Delta$ , corresponding to the permutations of the monomials $Z_{0}^{n+2},\ldots Z_{n+1}^{n+2}$ . This can be used to our advantage.

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Notation. Denote the vertices of $\partial\Delta_{\lambda}^{\vee}$ as $w_{0},\ldots,w_{n+1}$ , which coincide with the outward normal vectors because $\partial\Delta_{\lambda}^{\vee}=-\partial\Delta^{\vee}$ . Denote the vertices of $\Delta$ as $m^{0},\ldots,m^{n+1}$ , so that

\langle w_{i},m^{j}\rangle=\begin{cases}1,\quad&i\neq j,\\ -(n+1),\quad&i=j.\end{cases}

Let $\text{Star}(w_{i})$ be the star of $w_{i}$ in the barycentric subdivision of $\partial\Delta_{\lambda}^{\vee}$ . Let $Sing\subset\widetilde{Sing}$ be the subset of points not contained in the interior of any of these stars. The affine structure on $\partial\Delta_{\lambda}^{\vee}\setminus\widetilde{Sing}$ extends to $\partial\Delta_{\lambda}^{\vee}\setminus Sing$ , by decreeing that on the interior of $\text{Star}(w_{i})$ we use the coordinates for the chart $U_{w}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$ . As $Sing$ has codimension two inside $\partial\Delta_{\lambda}^{\vee}$ , this makes $\partial\Delta_{\lambda}^{\vee}$ into a singular affine manifold.

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Proposition 3.27. In the Fermat case, if $u$ is a locally convex function on $\partial\Delta_{\lambda}^{\vee}$ , which is invariant under the permutation group. Then $u$ satisfies the extension property.

00R5

Proof. We need to prove the characterisation in Prop. 3.19. Without loss of generality $L_{\lambda}(x)$ is achieved by $\langle m^{0},x\rangle+\lambda(m^{0})$ . We need to find $p\in\Delta$ , such that $u(y)-u(x)\geq\langle p,y-x\rangle.$ For this we study the gradient of the function $u_{m^{0}}$ on the various $w$ -charts.

First, notice for $x^{\prime},y^{\prime}$ on the face $\{L_{\lambda}=\langle m^{0},\rangle+\lambda(m^{0})\}$ , namely the convex hull of $w_{1},\ldots w_{n+1}$ , the vector $y^{\prime}-x^{\prime}$ is parallel to the face, and by convexity of $u_{m^{0}}$ the directional derivative $\nabla u_{m^{0}}\cdot(y^{\prime}-x^{\prime})$ is monotone along the path from $x^{\prime}$ to $y^{\prime}$ , so must be maximized at $y^{\prime}$ . In particular we consider such line segments on the face parallel to $w_{i}-w_{j}$ for $i,j\geq 1$ . By the discrete symmetry, $\nabla u_{m^{0}}\cdot(w_{j}-w_{i})$ must be zero on the plane of reflection bisecting the face. Thus for $i,j\geq 1$ , $i\neq j$ , the subset of the face

\{\nabla u_{m^{0}}\cdot w_{i}\geq\nabla u_{m^{0}}\cdot w_{j}\}\cap\{L_{\lambda}=\langle m^{0},\rangle+\lambda(m^{0})\}

agrees exactly with the half of the face containing $w_{i}$ . Therefore the subset of face

\{\nabla u_{m^{0}}\cdot w_{i}\geq\nabla u_{m^{0}}\cdot w_{j},\forall j\geq 1\}

is exactly the intersection of $\text{Star}(w_{i})$ with the face. Without loss of generality $x$ lies in $\text{Star}(w_{1})$ .

We follow the notation in the proof of Prop. 3.26. In the $w_{1}$ -chart, denote the gradient of $u_{m^{0}}$ as $\vec{p}$ , so that for $y$ in the $w_{1}$ -chart,

u_{m^{0}}(y)-u_{m^{0}}(x)\geq\vec{p}\cdot(y-x)_{w_{1}}.

A priori $\vec{p}$ lives in $M_{\mathbb{R}}/\mathbb{R}m^{0}$ . We lift $\vec{p}$ to $M_{\mathbb{R}}$ by demanding $\langle\vec{p},w_{1}\rangle=0$ , so by the above discussion $\langle\vec{p},w_{i}\rangle\leq 0$ for $i\geq 1.$ Define $p=\vec{p}+m_{0}$ , then $\langle p,w_{i}\rangle\leq 1$ for all $i\geq 1$ . We regard $p\in M_{\mathbb{R}}$ as the gradient of $u$ at $x$ , and write $p=\nabla u$ as a function of $x$ . This construction can be made on other faces as well, and on the intersection of two faces the definitions are compatible.

We claim $p\in\Delta$ : it suffices to show $\langle p,w_{0}\rangle\leq 1$ . Notice $w_{0}=-\sum_{1}^{n+1}w_{i}=\sum_{i=2}^{n+1}(w_{1}-w_{i})-(n+1)w_{1}$ . Consider the line segment in the face joining $x$ to the boundary of the face in the direction $\sum_{i=2}^{n+1}(w_{1}-w_{i})$ , which stays inside $\text{Star}(w_{1})$ , and along which $\nabla u\cdot\sum_{i=2}^{n+1}(w_{1}-w_{i})$ increases, or equivalently $\langle\nabla u,w_{0}\rangle$ increases. But the boundary of the face $\{L_{\lambda}=\langle m^{0},\rangle+\lambda(m^{0})\}$ lies also on a different face, and we can use the information from this new face to deduce $\langle\nabla u,w_{0}\rangle\leq 1$ there.

By construction for $y$ in the $w_{1}$ -chart,

u(y)-u(x)\geq\langle\vec{p}(x),(y-x)_{w_{1}}\rangle+\langle m_{0},y-x\rangle=\langle\nabla u(x),y-x\rangle.

We claim that in fact $u(y)-u(x)\geq\langle\nabla u(x),y-x\rangle$ holds for all $y\in\partial\Delta_{\lambda}^{\vee}$ . We are left to check for $y$ on the face $\{L_{\lambda}=\langle m^{1},\rangle+\lambda(m^{1})\}$ , namely the complement of the $w_{1}$ -chart. Consider the $w_{i}$ -chart for $i>1$ . We can write according to the decomposition $N_{\mathbb{R}}=(m^{1})^{\perp}\oplus\mathbb{R}w_{i}$ , that

y-x=(y-x)_{w_{i},m^{1}}+\langle m^{1},y-x\rangle w_{i}.

By local convexity, in the $w_{i}$ -chart $u_{m^{1}}$ is convex, so there is some $\vec{p}^{\prime}$ , such that for any $y^{\prime}$ in the $w_{i}$ -chart

u_{m^{1}}(y^{\prime})-u_{m^{1}}(x)\geq\vec{p}^{\prime}\cdot(y^{\prime}-x)_{w_{i},m^{1}}.

But a gradient vector of $u_{m^{1}}$ at $x$ is $\nabla u(x)-m^{1}$ , so we may take $\vec{p}^{\prime}=\nabla u(x)-m^{1}$ . Thus

u(y)-u(x)\geq\vec{p}^{\prime}\cdot(y-x)_{w_{i},m^{1}}+\langle m^{1},y-x\rangle=\langle\nabla u(x),y-x\rangle-\langle\vec{p}^{\prime},w_{i}\rangle\langle m^{1},y-x\rangle.

Now $\langle m^{1},y-x\rangle\geq 0$ as in the proof of Prop. 3.26, and $\langle\vec{p}^{\prime},w_{i}\rangle\leq 0$ by $\nabla u\in\Delta$ . This implies $u(y)-u(x)\geq\langle\nabla u(x),y-x\rangle$ as required.

We have verified the characterisation in Prop. 3.19, hence the extension property. ∎

The proof above contains some additional information about the gradients.

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Corollary 3.28. In the region $\text{Star}(w_{i})+\mathbb{R}_{\geq 0}w_{i}\subset N_{\mathbb{R}}$ , the directional derivative of the canonical extension $u=u^{**}$ satisfies $\langle w_{i},\nabla u\rangle=1.$ In particular, in this region, for any $m$ with $\langle m,w_{i}\rangle=1$ , the function $u_{m}=u-m$ is constant upon translation in the $w_{i}$ -direction.

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Proof. By Remark 3.21, the $\nabla u$ introduced in the above proof is actually the gradient of the extension $u$ over $N_{\mathbb{R}}$ . By the proof above, we know $\langle\nabla u,w_{i}\rangle=1$ on $\text{Star}(w_{i})\subset\partial\Delta_{\lambda}^{\vee}$ . This directional derivative can only increase as $x\in N_{\mathbb{R}}$ moves in the $w_{1}$ -direction. But $\nabla u\in\Delta$ on $N_{\mathbb{R}}$ since the extension is admissible, so $\langle\nabla u,w_{i}\rangle\leq 1$ everywhere, hence the claim. ∎

For later use, we define the notion of real MA equation in the Fermat case.

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Definition 3.29. Let $u$ be a locally convex function on $\partial\Delta_{\lambda}^{\vee}$ invariant under the discrete symmetry. Then $u$ is called an Aleksandrov solution of the real MA equation on $\partial\Delta_{\lambda}^{\vee}\setminus Sing$ if

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On the interior of any top dimensional face of $\partial\Delta_{\lambda}^{\vee}$ , in a set of standard local affine coordinates $x^{m_{1}},\ldots x^{m_{n}}$ with $dx^{m_{1}}\wedge dx^{m_{2}}\ldots dx^{m_{n}}$ equal to the standard volume form $d\mu_{\infty}$ , the function $u$ satisfies $MA(u)=d\mu_{\infty}$ in the Aleksandrov sense.
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On $\text{Star}(w)\subset U_{w}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$ , we use the standard affine coordinates $x^{m_{1}},\ldots x^{m_{n}}$ associated to the $U_{w}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$ chart. We demand for any vertex $m$ of $\Delta$ with $\langle m,w\rangle$ , the local function $u_{m}=u-m$ satisfies $MA(u_{m})=d\mu_{\infty}$ in the Aleksandrov sense.

Schematically we write $MA(u)=d\mu_{\infty}$ .

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Remark 3.30. Notice that the definition is compatible on overlapping charts because the transition functions lie in $SL(n,\mathbb{Z})\ltimes\mathbb{R}^{n}$ . On the locus $Sing\subset\partial\Delta_{\lambda}^{\vee}$ we make no definition.