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Mahler measures and logarithmic equidistribution. (Mesures de Mahler et équidistribution logarithmique.) (French) Zbl 1192.14020
Let \(X\) be an integral projective scheme defined over a number field \(F\) and \(v\) be a place of \(F\). Given a generic sequence \((x_n)_{n\geqslant 1}\) of algebraic points of \(X\), the equidistribution problem asks for the conditions under which the sequence of measures \((\eta_n)_{n\geqslant 1}\) defined by the average on the Galois orbit of \(x_n\) in the analytic space \(X_v^{\mathrm{an}}\) (in the sense of Berkovich if \(v\) is finite) converges weakly. In the Arakelov geometry approach of this problem, the appropriate condition is that the sequence \((x_n)_{n\geqslant 1}\) is small with respect to a suitable adelic line bundle \(\overline L\) on \(X\). That is, the sequence of heights \((h_{\overline L}(x_n))_{n\geqslant 1}\) converges to the normalized height of \(X\).
In the article under review, the authors consider a logarithmic variant of the equidistribution problem: given an adelic line bundle \(\overline M\) on \(X\) an a non-zero global section \(s\) of \(M\), does the sequence of integrals \((\int_{X_v^{\mathrm{an}}}\log\|s\|_v\,\mathrm{d}\eta_n)_{n\geqslant 1}\) converge? Note that this does not follow from the weak convergence of \((\eta_n)_{n\geqslant 1}\) since the function \(\log\|s\|_v\) many take the value \(-\infty\) on \(X_v^{\mathrm{an}}\). A conter-example is given in the article to show that the sequence of integrals need not converge in general. The authors prove that, if the normalized height of \(\mathrm{div}(s)\) coincides with that of \(X\), then the logarithmic equidistribution holds. They also establish a similar result for the logarithmic equidistribution of subvarieties of \(X\).
Reviewer: Huayi Chen (Paris)

MSC:
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14G22 Rigid analytic geometry
32U05 Plurisubharmonic functions and generalizations
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