A non-archimedean analogue of the Calabi-Yau theorem for totally degenerate abelian varieties (Q410091)

Let \(X\) be a compact complex manifold with an ample line bundle \(L\). It is known, by the Calabi-Yau theorem, that, for any smooth positive measure \(\mu\) on \(X\) with \(\int_X \mu = \int_X c_1 (L) ^{\wedge \dim (X)}\), then there exists a positive metric \(\| \, \|\) on \(L\), unique up to a constant multiple, such that \(c_1 (L, \| \, \|)^{\wedge \dim X} = \mu\).NEWLINENEWLINEIn the analogous situation of a non-Archimedean analytic space or a Berkovich analytic space [\textit{V. G. Berkovich}, Spectral theory and analytic geometry over non-Archimedean field. Mathematical Surveys and Monographs, 33. Providence, RI: American Mathematical Society (AMS) (1990; Zbl 0715.14013)] over \({\mathbb C}_p\), the same property is in general not anymore true (see for instance [\textit{A. Chambert-Loir} and \textit{A. Thuillier}, Ann. Inst. Fourier 59, No. 3, 977--1014 (2009; Zbl 1192.14020)].NEWLINENEWLINEIn the present paper the author shows an example of a non-Archimdean version of the existence part of the Calabi-Yau theorem. He considers a totally degenerate abelian variety \(A\) over \({\mathbb C}_p\) and a measure that is given as the direct image of a measure on a real skeleton of \(A\).