4.5 Legendre transform, extension, regularisation [00TR]

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4.5 Legendre transform, extension, regularisation

We restrict to the Fermat case, and consider a general $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$ with $\sup_{X_{s}}\varphi=0$ , invariant under the symmetric group permuting the monomials $Z_{0}^{n+2},\ldots,Z_{n+1}^{n+2}$ . The goal of this section is to canonically patch together the local convex functions in section 4.4 approximately to produce a convex admissible function on $N_{\mathbb{R}}=\mathbb{R}^{n+1}$ . We will then induce a potential $\psi\in PSH(X_{s},s^{-1}\omega_{FS})\cap C^{0}$ which is a regularisation of $\varphi$ in the sense that it enjoys better a priori bounds than $\varphi$ .

Proposition 4.14.

There is an admissible convex function $u$ on $N_{\mathbb{R}}$ , such that on $U_{w}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$ ,

|u-(\bar{\phi}_{m,w}+m)|\leq Cs^{-1/2}.

(24)

Proof.

The idea is to regard $\bar{\phi}_{m,w}+\langle m,x\rangle$ as approximately defining a locally convex function on $\partial\Delta_{\lambda}^{\vee}$ in the sense of Def. 3.22, and then the problem is essentially to prove an effective version of the extension property (cf. Prop. 3.27). We will outline the main modifications.

We will produce $u$ by mimicking the Legendre duality construction in Prop. 3.19. For $p\in\Delta$ , define

u^{*}(p)=\sup_{x\in\partial\Delta_{\lambda}^{\vee}}\{\langle x,p\rangle-(\bar{\phi}_{m,w}+\langle m,x\rangle)\},

where it is tacitly understood that $\bar{\phi}_{m,w}+\langle m,x\rangle$ is defined only over $\partial\Delta_{\lambda}^{\vee}\cap U^{\infty}_{w}$ , and the sup is taken over all choices of $m,w$ whenever $\bar{\phi}_{m,w}$ is defined. Since $\bar{\phi}_{m,w}$ are uniformly bounded on $\partial\Delta_{\lambda}^{\vee}$ , we see $\left\lVert u^{*}\right\rVert_{C^{0}(\Delta)}\leq C$ . We then define a convex function $u$ on $N_{\mathbb{R}}$ by another Legendre transform

u(x)=\sup_{p\in\Delta}\{\langle p,x\rangle-u^{*}(p)\},

which is admissible because $u^{*}$ is bounded. By the same reasoning in Prop. 3.19, on $\partial\Delta_{\lambda}^{\vee}\cap U^{\infty}_{w}$ ,

u(x)\leq\bar{\phi}_{m,w}+\langle m,x\rangle+Cs^{-1/2}.

We are only left to show

u(x)\geq\bar{\phi}_{m,w}+\langle m,x\rangle-Cs^{-1/2},

which amounts to showing that there exists $p\in\Delta$ , such that for any $y\in\partial\Delta_{\lambda}^{\vee}$ ,

\bar{\phi}_{m^{\prime},w^{\prime}}(y)+\langle m^{\prime},y\rangle\geq\bar{\phi}_{m,w}(x)+\langle m,x\rangle+\langle p,y-x\rangle-Cs^{-1/2}.

Notice our setting enjoys the discrete symmetry. This last step is the effective version of Prop. 3.27, and the proof is basically the same. ∎

By construction $u$ has a number of additional properties:

Corollary 4.15.

The canonical extension $u$ satisfies an a priori Lipschitz bound

\begin{cases}|u-\max_{m}\langle m,x\rangle|\leq C,\quad\forall x\in N_{\mathbb{R}},\\ |u(x)-u(x^{\prime})|\leq C|x-x^{\prime}|,\quad\forall x,x^{\prime}\in N_{\mathbb{R}}.\end{cases}

(25)

Morever, in the region $\text{Star}(w)+\mathbb{R}_{\geq 0}w\subset N_{\mathbb{R}}$ , for any $m$ with $\langle m,w\rangle=1$ , the function $u_{m}=u-m$ is constant upon translation in the $w$ -direction.

Proof.

The first inequality is because the Legendre transform $u^{*}(p)$ is bounded on $\Delta$ as in the above proof, and the second is because $\nabla u\in\Delta$ . The morever statement is essentially identical to Cor. 3.28. ∎

By a small variant of Prop. 3.16, when we pullback the admissible convex functions $u$ via $\text{Log}_{s}$ , we obtain a torus invariant Kähler current on $(\mathbb{P}_{\Delta},s^{-1}[\Delta])$ with continuous local potentials. In details, we write $\psi_{0}=u\circ\text{Log}_{s}$ , and define

\begin{cases}\psi=\psi_{0}-\frac{(n+2)}{2s}\log(\sum_{m^{\prime}\in vertex(\Delta)}e^{\frac{2}{n+2}\langle m^{\prime},\text{Log}(z)\rangle}),\\ \psi_{m}=\psi_{0}-\langle m,\text{Log}_{s}(z)\rangle=u_{m}\circ\text{Log}_{s}.\end{cases}

(26)

By construction $\psi\in PSH(\mathbb{P}_{\Delta},s^{-1}\omega_{FS})\cap C^{0}$ , and $\psi_{0},\psi_{m}$ are the local potentials of $\psi$ (cf. (19)). By Cor. 4.15, $\left\lVert\psi\right\rVert_{C^{0}}\leq C$ , and $\psi$ inherits the Lipschitz bound from $u$ . By a slight abuse of notation, the restriction to $X_{s}$ will still be denoted as $\psi\in PSH(X_{s},s^{-1}\omega_{FS})\cap C^{0}$ . We think of $\psi$ as a regularisation of $\varphi$ .

Remark 4.16.

As explained in section 2.3, on toric manifolds the Legendre transform arises from a limiting version of approximation by algebraic metrics, which in turn is a more standard way to regularise an arbitrary Kähler potential. Now $X_{s}$ is not a toric manifold, but the toric symmetry holds approximately in generic regions, which motivates us to take the Legendre transform as a replacement of algebraic regularisation.

We now specify some subregions on $X_{s}^{toric}$ with coordinate descriptions. These are intimately related to $\partial\Delta_{\lambda}^{\vee}\setminus Sing$ , which is covered by the stars of the vertices and the interior of the top dimensional faces (cf. section 3.5).

Notation.

(Star type regions on $X_{s}$ ) On the region $U^{s}_{w}\subset X_{s}$ , recall the coodinates $z^{m_{i}}$ and regard $x^{m_{i}}=s^{-1}\log|z^{m_{i}}|$ as local coordinates also on $U^{\infty}_{w}\cap\partial\Delta_{\lambda}^{\vee}$ . Let $U^{s,*}_{w}\subset U^{s}_{w,\delta}$ be the subset where the $x^{m_{i}}$ coordinates correspond to points in $\text{Star}(w)\subset\partial\Delta_{\lambda}^{\vee}$ . The tropical analogue of $U^{s,*}_{w}$ is $(\text{Star}(w)+\mathbb{R}_{\geq 0}w)\cap A_{\lambda}^{\infty}$ .

Notation.

(Face type regions on $X_{s}$ ) Consider a slightly shrinked subset of the interior of a given top dimensional face of $\partial\Delta_{\lambda}^{\vee}$ . This can be regarded as a subset of $U_{w,\delta}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$ , where we regard $x^{m_{i}}=s^{-1}\log|z^{m_{i}}|$ as local affine coordinates. Let $U_{w}^{s,face}\subset U_{w}^{s}$ be the subset where the $x^{m_{i}}$ coordinates correspond to points in this shrinked face. The tropical analogue of $U_{w}^{s,face}\subset U_{w}^{s}$ is the shrinked face.

The intuition is that when $z\in X_{s}$ has $\text{Log}_{s}$ image close to $\partial\Delta_{\lambda}^{\vee}$ , or if this image approaches infinity in specific directions, then $\varphi-\psi$ is bounded above by a very small number:

Proposition 4.17.

(Local potential upper bound)

•

Inside $U^{s,*}_{w}\subset X_{s}$ , for $\langle m,w\rangle=1$ , the local potentials satisfy $\varphi_{m}-\psi_{m}\leq Cs^{-1/2},$ or equivalently $\varphi-\psi\leq Cs^{-1/2}$ .
•

Inside $U^{s,face}_{w}$ , the local potentials satisfies $\varphi_{0}-\psi_{0}\leq Cs^{-1/2}$ , or equivalently $\varphi-\psi\leq Cs^{-1/2}$ .

Proof.

In the star type region case, by Cor. 4.7, we have the upper bound $\varphi_{m}-\bar{\phi}_{m,w}\leq Cs^{-1/2}$ . By Prop. 4.14 and Cor. 4.15, in $U^{s,*}_{w}$ we can replace $\bar{\phi}_{m,w}$ by $\psi_{m}$ up to an error bounded by $Cs^{-1/2}$ , hence the claim. The face type region follows the same argument, without the translational invariance statement of Cor. 4.15. ∎