Tamagawa numbers of polarized algebraic varieties

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Let L = (L, · ᵥ) be an ample metrized invertible sheaf on a smooth quasi-projective algebraic variety V defined over a number field. Denote by N(V, L,B) the number of rational points in V having L-height ≤ B. 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 τ L(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.