4.1 Harnack inequality [00TM]

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4.1 Harnack inequality

Consider a general possibly singular Kähler potential $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$ on $X_{s}$ , normalised to $\sup_{X_{s}}\varphi=0$ . We think of $\varphi$ equivalently as a collection of local potentials $\{\varphi_{0},\varphi_{m}\}$ as in section 3.3. 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. Recall $d\mu_{s}$ is the normalised canonical measure induced by the holomorphic volume form.

Notation.

Denote $X^{toric}_{s}$ as the union of all the toric regions $U_{w,\delta}^{s}$ for various choices of $m$ and $w$ . It is tacitly understood that slightly shrinked domains correspond to a slightly larger choice of $\delta$ , and we shall abusively use the same notation for shrinked domains.

Proposition 4.1.

(Harnack type inequality) Suppose $\varphi\in PSH(X_{s},s^{-1}\omega_{FS})$ with $\sup_{X_{s}}\varphi=0$ . Then the average integral

\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{X^{toric}_{s}}|\varphi|d\mu_{s}\leq C.

Proof.

(cf. proof of Prop. 3.1 in [2]) Consider the local potentials $\phi=\varphi_{m}$ on various coordinate charts in section 3.1, both of the toric type and of the boundary type. The charts can be chosen so that the Lebesgue measures thereof are uniformly equivalent to $d\mu_{s}$ up to a scaling factor. We have $|\phi-\varphi|\leq C$ uniformly on charts. Suppose a coordinate ball $B(p,3R)$ is contained in (the universal cover of) the local chart. Since $\phi$ is psh and $\phi-C\leq 0$ , for $z\in B(p,R)$ ,

\phi(y)-C\leq\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(y,2R)}(\phi-C)\lesssim\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(p,R)}(\phi-C),

hence

\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(p,R)}|\varphi|\lesssim 1+\inf_{B(p,R)}(-\varphi).

To deduce the global version of the Harnack type inequality we need a transitivity property, namely we can connect the chart containing the maximum point of $\varphi$ to any of the toric charts in $X_{s}^{toric}$ via a chain of $O(1)$ number of charts, such that $\inf_{B(p,R)}|\varphi|$ on charts increase by only $O(1)$ in each step. This last fact is because we can choose the chains of successive charts $B(p_{i},5R_{i})$ such that the measure of the overlap occupies a nontrivial portion of the previous chart:

|B(p_{i},R_{i})\cap B(p_{i+1},R_{i+1})|\gtrsim\frac{1}{10}|B(p_{i},R_{i})|,

which would force

\begin{split}\inf_{B(p_{i+1},R_{i+1})}|\varphi|\leq&\inf_{B(p_{i+1},R_{i+1})\cap B(p_{i},R_{i})}|\varphi|\leq\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(p_{i+1},R_{i+1})\cap B(p_{i},R_{i})}|\varphi|\\ \lesssim&\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.11674pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.48965pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.31259pt}}\!\int_{B(p_{i},R_{i})}|\varphi|\lesssim 1+\inf_{B(p_{i},R_{i})}|\varphi|.\end{split}

∎

Remark 4.2.

Notice this transitivity argument allows us to move from boundary type charts into toric charts, but not conversely, because the measure is much larger on toric charts.