Inventiones mathematicae

, Volume 200, Issue 2, pp 513–583 | Cite as

Seshadri constants, diophantine approximation, and Roth’s theorem for arbitrary varieties

Article

Abstract

In this paper, we associate an invariant \(\alpha _{x}(L)\) to an algebraic point \(x\) on an algebraic variety \(X\) with an ample line bundle \(L\). The invariant \(\alpha \) measures how well \(x\) can be approximated by rational points on \(X\), with respect to the height function associated to \(L\). We show that this invariant is closely related to the Seshadri constant \(\epsilon _{x}(L)\) measuring local positivity of \(L\) at \(x\), and in particular that Roth’s theorem on \(\mathbb {P}^1\) generalizes as an inequality between these two invariants valid for arbitrary projective varieties.

Mathematics Subject Classification

Primary 14G05 Secondary 14G40 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of Mathematics and StatisticsQueens UniversityKingstonCanada

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