Boundary asymptotics of the relative Bergman kernel metric for elliptic curves. II: Subleading terms (Q2836005)

Let \((X_{\lambda})_{\lambda\in\mathbb C\setminus\{0,1\}}\) be the so-called Legendre family of elliptic curves, where NEWLINE\[NEWLINE X_{\lambda}=\big\{(x,y)\in\mathbb C^2:y^2=x(x-1)(x-\lambda)\big\}\cup\big\{\infty\big\}, NEWLINE\]NEWLINE and let \(B_{\lambda}=k_{\lambda}dz\wedge d\bar{z}\) be the Bergman kernel of \(X_{\lambda}\) written in some local coordinate \(z\) for some local function \(k_{\lambda}\). If the fiber \(X_{\lambda}\) is smooth, then NEWLINE\[NEWLINE L_{\lambda,z}:=\sqrt{-1}\partial_{\lambda}\overline{\partial}_{\lambda}\log k_{\lambda}(z)\geq0. NEWLINE\]NEWLINE The main results of the paper under review are the following.NEWLINENEWLINE1. As \(\lambda\longrightarrow0\), then NEWLINE\[NEWLINE L_{\lambda,z}=\frac{\sqrt{-1}d\lambda\wedge d\bar{\lambda}}{|\lambda|^2(-\log|\lambda|^2)^2}\Bigg(1+\frac{2\log16}{\log|\lambda|}+3\bigg(\frac{\log16}{\log|\lambda|}\bigg)^2+4\bigg(\frac{\log16}{\log|\lambda|}\bigg)^3+O\bigg(\frac{1}{(\log|\lambda|)^4}\bigg)\Bigg). NEWLINE\]NEWLINENEWLINENEWLINE2. \(L_{\lambda,z}\) is the Poincaré metric of \(\mathbb C\setminus\{0,1\}\).NEWLINENEWLINE3. Let \(\omega_{0,1}\) denote the Poincaré metric of \(\mathbb C\setminus\{0,1\}\) and let \(p(\lambda)\) be its Kähler potential. Then, as \(\lambda\longrightarrow0\), NEWLINE\[NEWLINE p(\lambda)=-\log(-\log|\lambda|)+\log\pi+\frac{\log16}{\log|\lambda|}+O\bigg(\frac{1}{(\log|\lambda|)^2}\bigg). NEWLINE\]