Tamagawa numbers of polarized algebraic varieties

Abstract

Let $\mathcal{L}=(L,\Vert \cdot {\Vert }_v)$ be an ample metrized invertible sheaf on a smooth quasi-projective algebraic variety $V$ defined over a number field. Denote by $N(V,\mathcal{L},B)$ the number of rational points in $V$ having $\mathcal{L}$-height $\le B$. We consider the problem of a geometric and arithmetic interpretation of the asymptotic for $N(V,\mathcal{L},B)$ 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 $\mathcal{L}$-primitive varieties and $\mathcal{L}$-primitive fibrations. For $\mathcal{L}$-primitive varieties $V$ over $F$ we propose a method to define an adelic Tamagawa number ${\tau }_{\mathcal{L}}(V)$ 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 $N(V,\mathcal{L},B)$ on the choice of $v$-adic metrics on $\mathcal{L}$.

54 pages, minor corrections