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Continuity of volumes on arithmetic varieties. (English) Zbl 1167.14014
The notion of volume function is important in algebraic geometry. It is a birational invariant which measures the asymptotic behaviour of the sectional algebra of a line bundle on a projective variety. In the article under review, the author has studied the analogue of the volume function in Arakelov geometry, notably the continuity of the arithmetic volume function. Recall that the arithmetic volume function of a $$C^\infty$$ Hermitian line bundle $$\overline L$$ on an arithmetic variety $$X$$ of total dimension $$d$$ is defined as $\widehat{\mathrm{vol}}(\overline L):=\limsup_{m\rightarrow+\infty}\frac{\widehat{h}^0(X, m\overline L)}{m^d/d!},$ where $$\widehat{h}^0(X,m\overline L)$$ is defined as $$\log\#\{s\in H^0(X,L^{\otimes m})\,:\, \|s\|_{\sup}\leqslant 1\}$$. By the equality $$\widehat{\mathrm{vol}}(m\overline L)=m^{d}\widehat{\mathrm{vol}}(\overline L)$$, one can extend the definition domain of this function to $$\widehat{\mathrm{Pic}}(X)\otimes_{\mathbb Z}\mathbb Q$$, where $$\widehat{\mathrm{Pic}}(X)$$ denotes the group of smooth Hermitian line bundles on $$X$$, whose group law is written additively. Namely, for all $$\overline L,\overline A_1,\cdots,\overline A_n$$ in $$\widehat{\mathrm{Pic}}(X)$$, there exists a positive constant $$C$$ such that $\big|\widehat{\mathrm{vol}}(\overline L+\epsilon_1\overline A_1+\cdots+\epsilon_n\overline A_n)-\widehat{\mathrm{vol}}(\overline L)\big|\leqslant C\big(|\epsilon_1|+\cdots+|\epsilon_n|\big).$ The proof of this result uses an estimation of distorsion functions due to T. Bouche [Ann. Inst. Fourier 40, No. 1, 117–130 (1990; Zbl 0685.32015) and G. Tian [J. Differ. Geom. 32, No. 1, 99–130 (1990; Zbl 0706.53036)].
As an application, the author generalizes the arithmetic Hilbert-Samuel function to the nef case, in the following sense: assume that $$\overline L$$ and $$\overline N$$ are in $$\widehat{\mathrm{Pic}}(X)$$ with $$\overline L$$ nef, then $\widehat{h}^0(X,m\overline L+\overline N)=\frac{\widehat{c}_1(\overline L)^d}{d!}m^d+o(m^d)\qquad(m\rightarrow\infty).$ Consequently, a nef $$C^\infty$$ Hermitian line bundle $$\overline L$$ is big (i.e. $$\widehat{\mathrm{vol}}(\overline L)>0$$) if and only if $$\widehat{c}_1(\overline L)^d>0$$. Similarly, for $$C^\infty$$ Hermitian line bundle $$\overline L$$ with nef generic fibre and semi-positive metric, if $$\overline L$$ has moderate growth of positive even cohomologies, then the inequality $$\widehat{\mathrm{vol}}(\overline L)\geqslant \widehat{c}_1(\overline L)^d$$ holds. This result generalizes the arithmetic Hodge index theorem due to G. Faltings [Ann. Math. (2) 119, 387–424 (1984; Zbl 0559.14005)] and P. Hriljac [Am. J. Math. 107, 23–38 (1985; Zbl 0593.14004)] and the arithmetic Bogomolov-Gieseker’s inequality.
Reviewer: Huayi Chen (Paris)

##### MSC:
 14G40 Arithmetic varieties and schemes; Arakelov theory; heights
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