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On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties

  • Shu Kawaguchi EMAIL logo and Joseph H. Silverman
Published/Copyright: April 11, 2014
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Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2016 Issue 713

Abstract

Let f:X⤏X be a dominant rational map of a smooth projective variety defined over a characteristic 0 global field K, let δf be the dynamical degree of f, and let hX:X(K¯)→[1,∞) be a Weil height relative to an ample divisor. We prove that for every ϵ>0 there is a height bound

hX∘fn≪(δf+ϵ)nhX,

valid for all points whose f-orbit is well-defined, where the implied constant depends only on X, hX, f, and ϵ. An immediate corollary is a fundamental inequality α¯f(P)≤δf for the upper arithmetic degree. If further f is a morphism and D is a divisor satisfying an algebraic equivalence f*D≡βD for some β>δf, we prove that the canonical height

h^f,D=limβ-nhD∘fn

converges and satisfies h^f,D∘f=βh^f,D and h^f,D=hD+O(hX). We also prove that the arithmetic degree αf(P), if it exists, gives the main term in the height counting function for the f-orbit of P. We conjecture that α¯f(P)=δf whenever the f-orbit of P is Zariski dense and describe some cases for which we can prove our conjecture.

Funding source: JSPS grant-in-aid for young scientists (B)

Award Identifier / Grant number: 24740015

Funding source: NSF

Award Identifier / Grant number: DMS-0854755

Funding source: Simons Collaboration Grant

Award Identifier / Grant number: #241309

The authors would like to thank ICERM for providing a stimulating research environment during their spring 2012 visits, as well as the organizers of conferences on Automorphisms (Shirahama 2011), Algebraic Dynamics (Berkeley 2012), and the SzpiroFest (CUNY 2012), during which some of this research was done. The authors would also like to thank Najmuddin Fakhruddin and the referee for their helpful comments and suggestions regarding the initial version of this article, including pointing out that our original formulation of the main theorem was too general; see Remark 8 for details.

Received: 2012-12-21
Revised: 2014-1-24
Published Online: 2014-4-11
Published in Print: 2016-4-1

© 2016 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Congruence kernels around affine curves
  3. On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties
  4. On the splitting of quasilinear p-forms
  5. Graded quiver varieties and derived categories
  6. On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold
  7. Static Klein–Gordon–Maxwell–Proca systems in 4-dimensional closed manifolds. II
  8. Linear stability of algebraic Ricci solitons
  9. Dynamical stability of algebraic Ricci solitons
  10. Erratum to Dynamics of the Krichever construction in several variables (J. reine angew. Math. 572 (2004), 111–138)
  11. Addendum: The case of closed surfaces (Boundary value problems on planar graphs and flat surfaces with integer cone singularities I: The Dirichlet problem)
https://doi.org/10.1515/crelle-2014-0020

Articles in the same Issue

  1. Frontmatter
  2. Congruence kernels around affine curves
  3. On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties
  4. On the splitting of quasilinear p-forms
  5. Graded quiver varieties and derived categories
  6. On the limit of spectral measures associated to a test configuration of a polarized Kähler manifold
  7. Static Klein–Gordon–Maxwell–Proca systems in 4-dimensional closed manifolds. II
  8. Linear stability of algebraic Ricci solitons
  9. Dynamical stability of algebraic Ricci solitons
  10. Erratum to Dynamics of the Krichever construction in several variables (J. reine angew. Math. 572 (2004), 111–138)
  11. Addendum: The case of closed surfaces (Boundary value problems on planar graphs and flat surfaces with integer cone singularities I: The Dirichlet problem)
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