Tamagawa numbers of polarized algebraic varieties

TL;DR

This paper introduces a generalized notion of Tamagawa numbers for polarized algebraic varieties, linking geometric and arithmetic properties to asymptotic point counts and conjectures in the Minimal Model Program.

Contribution

It proposes a method to define adelic Tamagawa numbers for ${ m L}$-primitive varieties, extending Peyre's Tamagawa number to varieties with singularities and connecting these to point count asymptotics.

Findings

Tamagawa numbers depend on $v$-adic metrics on ${ m L}$.

Method applies to $Q$-Fano varieties with canonical singularities.

Examples illustrate the relation between Tamagawa numbers and asymptotic behavior.

Abstract

Let ${\cal L} = (L, \| \cdot \|_v)$ be an ample metrized invertible sheaf on a smooth quasi-projective algebraic variety $V$ defined over a number field. Denote by $N(V,{\cal L},B)$ the number of rational points in $V$ having ${\cal L}$-height $\leq B$. We consider the problem of a geometric and arithmetic interpretation of the asymptotic for $N(V,{\cal L},B)$ as $B \to \infty$ in connection with recent conjectures of Fujita concerning the Minimal Model Program for polarized algebraic varieties. We introduce the notions of ${\cal L}$-primitive varieties and ${\cal L}$-primitive fibrations. For ${\cal L}$-primitive varieties $V$ over $F$ we propose a method to define an adelic Tamagawa number $\tau_{\cal L}(V)$ which is a generalization of the Tamagawa number $\tau(V)$ introduced by Peyre for smooth Fano varieties. Our method allows us to construct Tamagawa numbers for $Q$-Fano varieties with at worst canonical singularities. In a series of examples of smooth polarized varieties and singular Fano varieties we show that our Tamagawa numbers express the dependence of the asymptotic of $N(V,{\cal L},B)$ on the choice of $v$-adic metrics on ${\cal L}$.