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Continuous extension of arithmetic volumes. (English) Zbl 1192.14022
Let $$X$$ be an integral projective flat scheme over $$\mathrm{Spec}(\mathbb Z)$$ and $$\overline{L}$$ be an invertible sheaf on $$X$$ equipped with a continuous metric on $$L_{\mathbb C}$$, invariant by the complex conjugation, the asymptotic behaviour of the spaces of small sections of $$\overline{L}^{\otimes n}$$ ($$n\geqslant 1$$) is described by the arithmetic volume function $$\widehat{\mathrm{vol}}(\overline L)$$. This function is the arithmetic analog of the volume function in projective algebraic geometry.
In the article under review, the author shows that the arithmetic volume function can be continuously extended to the arithmetic Picard group with coefficient in $$\mathbb R$$. This generalizes his previous result [J. Algebr. Geom. 18, No. 3, 407–457 (2009; Zbl 1167.14014)] on the continuity of the function $$\widehat{\mathrm{vol}}$$ on the arithmetic Picard group $$\widehat{\mathrm{Pic}}(X)$$. He also explain by an explicit example that the continuous extension does not follow formally from the continuity of $$\widehat{\mathrm{vol}}$$. The proof relies on a refinement of the main estimation on small sections established in [loc. cit.] by using the distorsion function.
Reviewer: Huayi Chen (Paris)

##### MSC:
 14G40 Arithmetic varieties and schemes; Arakelov theory; heights
##### Keywords:
arithmetic volume function; continuous extension
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