Inventiones mathematicae

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

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



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 



We thank Chris Dionne, Laurence Ein, Robert Lazarsfeld, Victor Lozovanu, and Damien Roy for helpful discussions. We are also extremely grateful to the referees of this paper for pointing out several mathematical and expositional errors in the initial versions, and for their suggestions on how to correct them. Finally, we wish to acknowledge an intellectual debt to Michael Nakamaye who has long advocated the point of view that Seshadri constants are diophantine.


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© 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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