4.2 Local potentials: convexity [00TN]

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4.2 Local potentials: convexity

We continue with a general $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$ normalised to $\sup_{X_{s}}\varphi=0$ , whose local potentials are $\{\varphi_{0},\varphi_{m}\}$ . A simple obeservation is:

Lemma 4.3.

Let $\Phi$ be any psh function on the open subset of $\{1<|\zeta_{i}|<\Lambda,i=1,\ldots n\}\subset(\mathbb{C}^{*})^{n}$ . Then the $T^{n}$ -invariant function

\bar{\Phi}(\log|\zeta_{1}|,\ldots,\log|\zeta_{n}|)=\frac{1}{(2\pi)^{n}}\int_{T^{n}}\Phi(|\zeta_{1}|e^{i\theta_{1}},\ldots|\zeta_{n}|e^{i\theta_{n}})d\theta_{1}\ldots d\theta_{n}

is a convex function in the variables $x_{1}=\log|\zeta_{1}|,\ldots,x_{n}=\log|\zeta_{n}|$ .

Proof.

Since the $T^{n}$ -action on $(\mathbb{C}^{*})^{n}$ is holomorphic, $\Phi(\zeta_{1}e^{i\theta_{1}},\ldots\zeta_{n}e^{i\theta_{n}})$ is psh in $\zeta$ for any choice of $\theta_{i}$ , so the average function $\bar{\Phi}$ is also psh. Any $T^{n}$ -invariant psh function must be convex in the log coordinates, because of the formula

\sqrt{-1}\partial\bar{\partial}\bar{\Phi}=\frac{1}{4}\sum\frac{\partial^{2}\bar{\Phi}}{\partial x_{i}\partial x_{j}}\sqrt{-1}d\log\zeta_{i}\wedge d\overline{\log\zeta_{j}}\geq 0.

∎

In the region $U_{w}^{s}\subset X_{s}$ , we can find $m\in\Delta_{\mathbb{Z}}$ with $\langle m,w\rangle=1$ and $\mathbb{C}^{*}$ -coordinates $z^{m_{1}},\ldots z^{m_{n}}$ as in section 3.1, and consider the local potential $\phi=\varphi_{m}$ . Denote $x^{m_{i}}=\frac{\log|z^{m_{i}}|}{s}$ . We produce the local average function

\bar{\phi}(x^{m_{1}},\ldots x^{m_{n}})=\frac{1}{(2\pi)^{n}}\int_{T^{n}}\phi(|z^{m_{1}}|e^{i\theta_{1}},\ldots|z^{m_{n}}|e^{i\theta_{n}})d\theta_{1}\ldots d\theta_{n}.

(22)

Proposition 4.4.

In the chart $U^{s}_{w}$ the average function $\bar{\phi}$ is convex, and on the shrinked chart $U^{s}_{w,\delta}$ it has a Lipschitz bound:

|\bar{\phi}|\leq C,\quad|\bar{\phi}(x)-\bar{\phi}(x^{\prime})|\leq C|x-x^{\prime}|.

(23)

Proof.

By Lemma 4.3, $\bar{\phi}$ is convex, and by Prop. 4.1 it has an $L^{1}$ bound in the $x^{m_{i}}$ coordinates:

\int|\bar{\phi}|dx^{m_{1}}\ldots dx^{m_{n}}\leq C.

Clearly $\bar{\phi}$ is also bounded above, so for the argument we may pretend $\bar{\phi}\leq 0$ upon shifting by a bounded constant.

We claim $\bar{\phi}(x)$ is bounded from below for $x$ in a shrinked interior region. The ball $B(x,2r)$ is contained in the coordinate chart, with $r$ bounded below by a positive constant. For $y$ in the annulus $B(x,2r)\setminus B(x,r)$ , we have $2\bar{\phi}(\frac{x+y}{2})\leq\bar{\phi}(x)+\bar{\phi}(y)$ , so upon integration

\int|\bar{\phi}|\gtrsim\int 2|\bar{\phi}(\frac{x+y}{2})|dy\geq\int|\bar{\phi}(x)|+|\bar{\phi}(y)|dy,

which bounds $|\bar{\phi}(x)|$ . Thus on a slightly shrinked $x$ -domain the oscillation is bounded:

\text{osc }\bar{\phi}=(\sup-\inf)\bar{\phi}\leq C,

and the Lipschitz bound follows again by convexity. ∎

Remark 4.5.

We discuss some intuition about log scales. Let $P\in X_{s}$ lie in $U_{w,\delta}^{s}$ , then a log scale $|z^{m_{i}}|\sim|z^{m_{i}}(P)|$ around $P$ refers to the subregion

\{\frac{1}{2}|z^{m_{i}}(P)|\lesssim|z^{m_{i}}|\lesssim 2|z^{m_{i}}(P)|,\quad 1\leq i\leq n\}.

Now $\log|z^{m_{i}}|$ vary by order $O(s)$ within $U_{w,\delta}^{s}$ , so there are an enormous number of log scales. The long range behaviour of $X_{s}$ is similar to $(\mathbb{C}^{*})^{n}$ , with half of the dimensions compactified into $T^{n}$ . On the other hand, over one log scale $X_{s}$ behaves qualitatively like the unit disc in $\mathbb{C}^{n}$ . The concept of local oscillation of a function refers to the oscillation within one log scale. In particular the Lipschitz bound (23) implies a local oscillation bound

\text{osc}_{|z^{m_{i}}|\sim|z^{m_{i}}(P)|}\bar{\phi}\leq Cs^{-1}.