Big line bundles over arithmetic varieties

By Xinyi Yuan. 2006

188 citations recorded by the source corpus.

We prove a Hilbert-Samuel type result of arithmetic big line bundles in Arakelov geometry, which is an analogue of a classical theorem of Siu. An application of this result gives equidistribution of small points over algebraic dynamical systems, following the work of Szpiro-Ullmo-Zhang. We also generalize Chambert-Loir's non-archimedean equidistribution.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/368407019d61a31a1b970c8f242183159714aa13

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Distribution of postcritically finite polynomials (2013, unknown): “hen Zm ∩ D= ∅ for all mlarge enough. It is called small if hL¯(Zm) := 1 Card(Z m) P x∈Z hL¯(x) tends to 0 as m→ ∞. We can now state the following slight generalization of Yuan’s theorem. Theorem 3.3 ([Y]). Let K be a number ﬁeld, X be a projective variety over K of dimension d−1, L→ Xan ample line b”
Towards a Dynamical Manin–Mumford Conjecture (2011, background): “This follows immediately from the fact that the equidistribution results of [2, 4, 7, 21] apply to such families of points.”
Formes différentielles réelles et courants sur les espaces de Berkovich (2012, unknown): “Elle donne aussi lieu à des théorèmes d’équidistribution des points de petite hauteur tout à fait analogues à ceux évoqués plus haut dans le cas archimédien ([22, 4, 36, 40, 58]).”
On volumes of arithmetic line bundles (2008, methodology): “They use different definitions of bigness, but [ Yu , Corollary 2.4] shows that all of these definitions are equivalent.”
Heights and measures on analytic spaces. A survey of recent results, and some remarks (2010, methodology): “The relevance of Berkovich spaces in Diophantine geometry has now made been clear by many papers; besides [19] and [20] and the general equidistribution theorem of Yuan [59], I would like to mention the works [38, 39, 40, 30] who discuss the function field case of the equidistribution theorem, as we”
Continuity of volumes on arithmetic varieties (2006, methodology): “The second way: In the paper [ 16 ], for a fixed smooth member Y ∈ |A|, Yuan considered an exact sequence”
Equidistribution in Families of Abelian Varieties and Uniformity (2021, unknown): “In addition, Yuan’s bigness theorem [93] implies (REC) in the case of non-archimedean places ν ∈ Σ f ( K ).”
Convex Bodies Associated to Linear Series (2008, background): “There has also been some very interesting recent work on arithmetic analogues of the volume function [43], [30], which leads to: Qɪɴ 7.”