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4.6 Improved Skoda inequality [00TS]

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4.6 Improved Skoda inequality

Recall the local L1L^{1}-oscillation bounds in both toric and boundary type regions, from Cor. 4.8 and 4.11. Consequently,

Lemma 4.18.

(Local Skoda estimate) Consider any φ∈P​S​H​(Xs,s−1​ωF​S)\varphi\in PSH(X_{s},s^{-1}\omega_{FS}) normalised to supXsφ=0\sup_{X_{s}}\varphi=0. There are uniform positive constants α\alpha, CC, such that the local potentials ϕ\phi satisfy

  • •

    In a log scale in the toric region,

    −∫|zmi|∼|zmi​(P)|e−α​s​(ϕ−−∫ϕ)dμs≤C.\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int_{|z^{m_{i}}|\sim|z^{m_{i}}(P)|}e^{-\alpha\sqrt{s}(\phi-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-6.57559pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-4.84631pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-3.71837pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-3.36836pt}}\!\int\phi)}d\mu_{s}\leq C.
  • •

    In a boundary type chart,

    −∫UPe−α​s​(ϕ−−∫ϕ)dμs≤C.\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int_{U_{P}}e^{-\alpha\sqrt{s}(\phi-\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-6.57559pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-4.84631pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-3.71837pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-3.36836pt}}\!\int\phi)}d\mu_{s}\leq C.
Proof.

Apply the standard Skoda inequality (cf. Thm 2.1) to the rescaled function s1/2​(ϕ−−∫ϕ)s^{1/2}(\phi-\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\phi). ∎

Remark 4.19.

The local average −∫ϕ\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\phi can be replaced by the local supremum using the mean value inequality.

Corollary 4.20.

(global Skoda estimate) Consider any φ∈P​S​H​(Xs,s−1​ωF​S)\varphi\in PSH(X_{s},s^{-1}\omega_{FS}) normalised to supXsφ=0\sup_{X_{s}}\varphi=0. There are uniform positive constants α\alpha, CC, such that

−∫Xse−α​φdμs≤C.\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_{s}}e^{-\alpha\varphi}d\mu_{s}\leq C. (27)
Proof.

By the local Skoda estimate and the Remark above, for both a log scale in the toric region, and a boundary type chart, the local average

−∫eα​s​(−φ+supl​o​cφ)dμs≤C,\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 e^{\alpha\sqrt{s}(-\varphi+\sup_{loc}\varphi)}d\mu_{s}\leq C, (28)

so in particular −∫eα⁡(−φ+supl​o​cφ)≤C.\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 e^{\alpha(-\varphi+\sup_{loc}\varphi)}\leq C. But we have already achieved a C0C^{0}-bound on local average functions, and in particular a lower bound on local suprema. Thus

−∫e−α​φdμs≤Ce−αsupl​o​cφ≤C,\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 e^{-\alpha\varphi}d\mu_{s}\leq Ce^{-\alpha\sup_{loc}\varphi}\leq C,

or equivalently ∫l​o​ce−α​φ​d​μs≤C​∫l​o​cd​μs\int_{loc}e^{-\alpha\varphi}d\mu_{s}\leq C\int_{loc}d\mu_{s} for local integrals. To pass from this to the global Skoda estimate, we need to take a large collection of log scales and boundary type charts and sum over the estimates:

∫Xse−α​φ​d​μs≤C​∑∫l​o​cd​μs.\int_{X_{s}}e^{-\alpha\varphi}d\mu_{s}\leq C\sum\int_{loc}d\mu_{s}.

The only problem is to ensure that the local charts can be chosen without substantially overcounting the measure. For points on XsX_{s} whose Logs\text{Log}_{s} image is at O⁡(s−1)O(s^{-1}) Euclidean distance to ∂Δλ∨\partial\Delta_{\lambda}^{\vee}, it is easy to choose the charts so that each point is contained in O⁡(1)O(1) number of charts. Away from ∂Δλ∨\partial\Delta_{\lambda}^{\vee}, the points deep inside the boundary type charts in general do not have this local finiteness property, but this is compensated by the fact that the measure d​μsd\mu_{s} decays exponentially away from ∂Δλ∨\partial\Delta_{\lambda}^{\vee} (cf. (16)). The conclusion is that

∑∫l​o​cd​μs≤C​∫Xsd​μs,\sum\int_{loc}d\mu_{s}\leq C\int_{X_{s}}d\mu_{s},

whence the global Skoda estimate. ∎

We now specialize to the Fermat case, and consider φ∈P​S​H​(X,s−1​ωF​S)\varphi\in PSH(X,s^{-1}\omega_{FS}) normalised to supXsφ=0\sup_{X_{s}}\varphi=0 with discrete symmetry, as in section 4.5. The regularisation of φ\varphi produced via Legendre transform is denoted as ψ\psi.

Theorem 4.21.

(Improved Skoda estimate) In the Fermat case above, there are uniform constants α\alpha, CC, such that

−∫Xse−α​s​(φ−ψ)dμs≤C.\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int_{X_{s}}e^{-\alpha\sqrt{s}(\varphi-\psi)}d\mu_{s}\leq C. (29)
Proof.

On either a log scale in the toric region, or a boundary type chart, we have by the local L1L^{1}-oscillation estimate and the mean value inequality that

|supl​o​cφm−−∫l​o​cφm|≤Cs−1/2,|supl​o​cψm−−∫l​o​cψm|≤Cs−1/2.|\sup_{loc}\varphi_{m}-\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_{loc}\varphi_{m}|\leq Cs^{-1/2},\quad|\sup_{loc}\psi_{m}-\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_{loc}\psi_{m}|\leq Cs^{-1/2}.

Notice also the local averages of φm\varphi_{m} and ψm\psi_{m} differ by O(s−1/2)O(s^{-1/2}), so

|supl​o​cφm−supl​o​cψm|≤Cs−1/2,|supl​o​cφ−supl​o​cψ|≤Cs−1/2.|\sup_{loc}\varphi_{m}-\sup_{loc}\psi_{m}|\leq Cs^{-1/2},\quad|\sup_{loc}\varphi-\sup_{loc}\psi|\leq Cs^{-1/2}.

Combined with (28),

−∫l​o​ce−α​s​(φ−ψ)dμs≤C.\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_{loc}e^{-\alpha\sqrt{s}(\varphi-\psi)}d\mu_{s}\leq C.

The summation argument as in the global Skoda estimate proves the claim. ∎

Remark 4.22.

This means ϕ−ψ\phi-\psi can only fail to be bounded below by Cs−1/2Cs^{-1/2} on a set with exponentially small probability measure. Notice we have not yet used the complex MA equation.

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