Tamagawa numbers of polarized algebraic varieties

Abstract

Let L = (L, ‖ · ‖v) be an ample metrized invertible sheaf on a smooth quasi-projective algebraic variety V over a number field F . Denote by N(V,L, B) the number of rational points in V having Lheight ≤ B. In this paper we consider the problem of a geometric and arithmetic interpretation of the asymptotic for N(V,L, B) as B → ∞ in connection with recent conjectures of Fujita concerning the Minimal Model Program for polarized algebraic varieties. We introduce the notions of L-primitive varieties and L-primitive fibrations. For L-primitive varieties V over F we propose a method to define an adelic Tamagawa number τL(V ) which is a generalization of the Tamagawa number τ(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,L, B) on the choice of v-adic metrics on L.

Locations

• arXiv (Cornell University) - View - PDF
• French digital mathematics library (Numdam) - View - PDF
• Astérisque - View