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 $\mathcal{L}$.

arXiv comment54 pages, minor corrections

References in corpus (5)

• Rational points on some Fano cubic bundles
• Rational Points of Bounded Height on Compactifications of Anisotropic Tori
• Equivariant resolution of singularities in characteristic 0
• Kodaira energies of polarized log varieties
• Notes on Kodaira energies of polarized varieties

Cited by in corpus (12)

• Rational points on compactifications of semi-simple algebraic groups
• Integral points of bounded height on partial equivariant compactifications of vector groups
• Manin's conjecture for quartic del Pezzo surfaces with a conic fibration
• Height zeta functions of equivariant compactifications of the Heisenberg group
• Campana points of bounded height on vector group compactifications
• Integral points on singular del Pezzo surfaces
• Varieties with too many rational points
• Manin's conjecture for certain spherical threefolds
• Integral points of bounded height on a certain toric variety
• The power-saving Manin-Peyre's conjectures for a senary cubic
• The geometric exceptional set in Manin's conjecture for Batyrev and Tschinkel's example
• Open problems in the wild McKay correspondence and related fields