[Paper Review] Continuity of volumes on arithmetic varieties

Atsushi Moriwaki|ArXiv.org|Dec 11, 2006

Algebraic Geometry and Number Theory14 references4 citations

TL;DR

This paper establishes the continuity of the arithmetic volume function for $C^{inity}$-hermitian invertible sheaves on arithmetic varieties, proving that the volume function is continuous on the rational Néron-Severi group. The key result is an arithmetic Hilbert-Samuel formula for nef hermitian line bundles, which characterizes bigness via positivity of the arithmetic self-intersection number.

ABSTRACT

We introduce the volume function for hermitian invertible sheaves on an arithmetic variety as an analogue of the geometric volume function. The main result of this paper is the continuity of the arithmetic volume function. As a consequence, we have the arithmetic Hilbert-Samuel formula for a nef hermitian invertible sheaf. We also give another several applications, for example, a generalized Hodge index theorem, an arithmetic Bogomolov-Gieseker's inequality, etc.

Motivation & Objective

• To define and study the arithmetic volume function for $C^{inity}$-hermitian invertible sheaves on arithmetic varieties as an analogue of the geometric volume function.
• To establish the continuity of the arithmetic volume function on the rational Néron-Severi group, resolving a fundamental question in arithmetic geometry.
• To derive the arithmetic Hilbert-Samuel formula for nef hermitian line bundles, linking bigness to positivity of arithmetic self-intersection numbers.
• To prove a generalized Hodge index theorem and an arithmetic Bogomolov-Gieseker inequality as applications of the continuity result.

Proposed method

• Define the arithmetic volume function via the limsup of logarithmic growth of sections with sup-norm bounded by 1.
• Prove homogeneity of the volume function and extend it to the rational Néron-Severi group using this property.
• Establish a key technical estimate (Theorem C) involving sections of line bundles on arithmetic varieties with controlled growth of cohomology.
• Use direct estimates based on the geometry of divisors and cohomological bounds to prove continuity, avoiding reliance on arithmetic Riemann-Roch.
• Apply the continuity result to derive the arithmetic Hilbert-Samuel formula and generalized Hodge index theorems under conditions of nefness and semipositivity.
• Use resolution of singularities and ampleness assumptions to reduce the problem to known cases and apply cohomological growth estimates.

Experimental results

Research questions

• RQ1 Is the arithmetic volume function continuous on the rational Néron-Severi group of an arithmetic variety?
• RQ2 Does the arithmetic Hilbert-Samuel formula hold for nef $C^{inity}$-hermitian line bundles?
• RQ3 Can a generalized Hodge index theorem be established for arithmetic varieties under conditions of nefness and semipositivity?
• RQ4 Under what conditions does positivity of the arithmetic self-intersection number imply bigness of a hermitian line bundle?
• RQ5 Can the arithmetic Bogomolov-Gieseker inequality be derived from the generalized Hodge index theorem?

Key findings

• The arithmetic volume function is continuous on $\widehat{\operatorname{Pic}}(X)\otimes\mathbb{Q}$, which implies the validity of the arithmetic Hilbert-Samuel formula for nef hermitian line bundles.
• For a nef $C^\infty$-hermitian line bundle $\overline{L}$, the arithmetic Hilbert-Samuel formula holds: $\log\#\{s\in H^0(X,mL+N)\mid\|s\|_{\sup}\leq 1\} = \frac{\widehat{\operatorname{deg}}(\widehat{c}_1(\overline{L})^{\cdot d})}{d!}m^d + o(m^d)$ as $m\to\infty$.
• The arithmetic volume satisfies $\widehat{\operatorname{vol}}(\overline{L}) = \widehat{\operatorname{deg}}(\widehat{c}_1(\overline{L})^{\cdot d})$ for nef $\overline{L}$, and $\overline{L}$ is big if and only if $\widehat{\operatorname{deg}}(\widehat{c}_1(\overline{L})^{\cdot d}) > 0$.
• A generalized Hodge index theorem holds: if $L_\mathbb{Q}$ is nef, $c_1(\overline{L})$ is semipositive, and $L$ has moderate cohomological growth, then $\widehat{\operatorname{vol}}(\overline{L}) \geq \widehat{\operatorname{deg}}(\widehat{c}_1(\overline{L})^{\cdot d})$.
• The arithmetic Bogomolov-Gieseker inequality holds for Einstein-Hermitian hermitian vector bundles on arithmetic surfaces: $\widehat{\operatorname{deg}}\left(\widehat{c}_2(\overline{E}) - \frac{r-1}{2r}\widehat{c}_1(\overline{E})^2\right) \geq 0$.
• The continuity of the volume function allows the conclusion that $\overline{L}$ is big whenever $\widehat{\operatorname{deg}}(\widehat{c}_1(\overline{L})^{\cdot d}) > 0$, under the conditions of nefness on fibers and semipositivity of the first Chern form.