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Relative dynamical degrees of correspondences over a field of arbitrary characteristic

  • Tuyen Trung Truong EMAIL logo
Veröffentlicht/Copyright: 10. Januar 2018
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Journal fĂŒr die reine und angewandte Mathematik (Crelles Journal)
Aus der Zeitschrift Journal fĂŒr die reine und angewandte Mathematik (Crelles Journal) Band 2020 Heft 758

Abstract

Let 𝕂 be an algebraically closed field of arbitrary characteristic, X and Y irreducible possibly singular algebraic varieties over 𝕂. Let f:X⊱X and g:Y⊱Y be dominant correspondences, and π:X⇱Y a dominant rational map which semi-conjugate f and g, i.e. so that π∘f=g∘π. We define relative dynamical degrees λp(f|π)≄1 for any p=0,
,dim⁥(X)-dim⁥(Y). These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when Y is smooth and g is a multiple of a rational map, and are birational invariants. More generally, a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semi-conjugacy (φ,ψ) from π2:(X2,f2)→(Y2,g2) to π1:(X1,f1)→(Y1,g1) we have λp(f1|π1)≄λp(f2|π2) for all p. Many of our results are new even when 𝕂=ℂ. Self-correspondences are abundant, even on varieties having not many self rational maps, hence these results can be applied in many situations. In the last section of the paper, we will discuss recent new applications of this to algebraic dynamics, in particular to pullback on l-adic cohomology groups in positive characteristics.

Funding source: Australian Research Council

Award Identifier / Grant number: DP120104110

Award Identifier / Grant number: DP150103442

Funding statement: The author was partially supported by Australian Research Council grants DP120104110 and DP150103442.

Acknowledgements

We would like to thank Finnur Larusson for suggesting using de Jong’s alteration, which is very crucial in the treatment of this paper, to him and Erik LĂžw for helping with the presentation of the paper. We thank Keiji Oguiso for checking thoroughly several earlier versions of this paper, his interest in the results of the paper and constant encouragement in the course of this work. We would also like to thank Tien-Cuong Dinh, HĂ©lĂšne Esnault, Charles Favre, Mattias Jonsson, Pierre Milman and Claire Voisin for their invaluable help. The discussion with Nguyen-Bac Dang on his paper [8] was also very helpful. We are also grateful to many comments and suggestions of the referees, which helped improve the paper, in particular for pointing out a gap in the original proof of Lemma 4.1 and for suggesting Lemma 3.2.

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Received: 2016-12-12
Revised: 2017-11-01
Published Online: 2018-01-10
Published in Print: 2020-01-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Limit sets of TeichmĂŒller geodesics with minimal nonuniquely ergodic vertical foliation, II
  3. On metrics on 2-orbifolds all of whose geodesics are closed
  4. The free-boundary Brakke flow
  5. Relative dynamical degrees of correspondences over a field of arbitrary characteristic
  6. Harmonic measure and Riesz transform in uniform and general domains
  7. Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type
  8. A proof of Milnor conjecture in dimension 3
  9. Round spheres are Hausdorff stable under small perturbation of entropy
  10. A local regularity theorem for mean curvature flow with triple edges
https://doi.org/10.1515/crelle-2017-0052

Artikel in diesem Heft

  1. Frontmatter
  2. Limit sets of TeichmĂŒller geodesics with minimal nonuniquely ergodic vertical foliation, II
  3. On metrics on 2-orbifolds all of whose geodesics are closed
  4. The free-boundary Brakke flow
  5. Relative dynamical degrees of correspondences over a field of arbitrary characteristic
  6. Harmonic measure and Riesz transform in uniform and general domains
  7. Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type
  8. A proof of Milnor conjecture in dimension 3
  9. Round spheres are Hausdorff stable under small perturbation of entropy
  10. A local regularity theorem for mean curvature flow with triple edges
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