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Dynamics on Berkovich Spaces in Low Dimensions

Part of the Lecture Notes in Mathematics book series (LNM,volume 2119)

Abstract

These are expanded lecture notes for the summer school on Berkovich spaces that took place at the Institut de Mathématiques de Jussieu, Paris, during June 28–July 9, 2010. They serve to illustrate some techniques and results from the dynamics on low-dimensional Berkovich spaces and to exhibit the structure of these spaces.

Keywords

  • Berkovich Spaces
  • Semivaluations
  • Exceptional Prime
  • Berkovich Projective Line
  • Integral Affine Structure

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    For a precise definition of what we mean by “tree”, see Sect. 2.

  2. 2.

    Its one-point compactification is the Berkovich projective line \(\mathop{\mathbf{P}_{\mathrm{Berk}}^{1}}\nolimits =\mathop{ \mathbf{A}_{\mathrm{Berk}}^{1}}\nolimits \cup \{\infty \}\).

  3. 3.

    In [FJ04, FJ07], the valuative tree is denoted by \(\mathcal{V}\). We write \(\mathcal{V}_{0}\) here in order to emphasize the choice of point 0 ∈ A 2.

  4. 4.

    The apex of the cone does not define an element in \(\mathop{\mathbf{A}_{\mathrm{Berk}}^{2}}\nolimits\).

  5. 5.

    In [FJ07, FJ11], the valuative tree at infinity is denoted by \(\mathcal{V}_{0}\), but the notation \(\mathcal{V}_{\infty }\) seems more natural.

  6. 6.

    In [BR10], the local degree is called multiplicity.

  7. 7.

    Our definition of “tree” is not the same as the one used in set theory [Jec03] but we trust that no confusion will occur. The terminology “R-tree” would have been natural, but has already been reserved [GH90] for slightly different objects.

  8. 8.

    Except for the missing condition (RT3), see Remark 2.5.

  9. 9.

    Unfortunately, the terminology is not uniform across the literature. In [BGR84, Ber90] ‘valuation’ is used to denoted multiplicative norms. In [FJ04], ‘valuation’ instead of ‘semi-valuation’ is used even when the prime ideal \(\{v = +\infty \}\) is nontrivial.

  10. 10.

    They are sometimes called rigid points as they are the points that show up rigid analytic geometry [BGR84].

  11. 11.

    This differs from the “algebraic degree” used by Trucco, see [Tru09, Definition 5.1].

  12. 12.

    The terminology “tame” follows Trucco [Tru09].

  13. 13.

    In [Bak09, BR10], the function − G is called the normalized Arakelov-Green’s function with respect to the Dirac mass at x 0.

  14. 14.

    The analytification of a general variety or scheme over K is defined by gluing the analytifications of open affine subsets, see [Ber90, §3.5].

  15. 15.

    The unit polydisc is denoted by E(0, 1) in [Ber90, §1.5.2].

  16. 16.

    The center map is called the reduction map in [Ber90, §2.4]. We use the valuative terminology center as in [Vaq00, §6] since it will be convenient to view the elements of \(\mathop{\mathbf{A}_{\mathrm{Berk}}^{n}}\nolimits\) as semivaluations rather than seminorms.

  17. 17.

    This is a divisorial valuation given by the order of vanishing along the exceptional divisor of the blowup of Y, see Sect. 6.10.

  18. 18.

    A local branch is a preimage of a point of C under the normalization map.

  19. 19.

    The acronym “nef” is due to M. Reid who meant it to stand for “numerically eventually free” although many authors refer to it as “numerically effective”.

  20. 20.

    A higher-dimensional version of this result is known as the “Negativity Lemma” in birational geometry: see [KM98, Lemma 3.39] and also [BdFF10, Proposition 2.11].

  21. 21.

    The log discrepancy is called thinness in [FJ04, FJ05a, FJ05b, FJ07].

  22. 22.

    We shall not, however, actually consider the toric variety defined by the fan \(\hat{\Delta }(\pi )\).

  23. 23.

    The increasing parametrization −α is denoted by α and called skewness in [FJ04]. The increasing parametrization A is called thinness in loc. cit..

  24. 24.

    The notation reflects the fact that \(\vert \cdot \vert:= e^{-v}\) is a seminorm on R, see (30).

  25. 25.

    If \(\varphi \in \mathop{ SH}\nolimits (\mathcal{V}_{0})\), then \(-\varphi\) is a positive tree potential in the sense of [FJ04].

  26. 26.

    The existence of such a relation has been established by W. Gignac and M. Ruggiero in arXiv:1209.3450.

  27. 27.

    The notation in these notes differs from [FJ07, FJ11] where the valuative tree at infinity is denoted by \(\mathcal{V}_{0}\). In loc. cit. the valuation \(\mathop{ord}\nolimits _{\infty }\) defined in (64) is denoted by \(-\deg\).

  28. 28.

    In [FJ04] the parametrization A is called thinness whereas −α is called skewness.

  29. 29.

    As in Sect. 7.8 the notation reflects the fact that \(\vert \cdot \vert:= e^{-v}\) is a seminorm on R.

  30. 30.

    P. Mondal has given examples in arXiv:1301.3172 showing that equality does not always hold.

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Acknowledgements

I would like to express my gratitude to many people, first and foremost to Charles Favre for a long and fruitful collaboration and without whom these notes would not exist. Likewise, I have benefitted enormously from working with Sébastien Boucksom. I thank Matt Baker for many interesting discussions; the book by Matt Baker and Robert Rumely has also served as an extremely useful reference for dynamics on the Berkovich projective line. I am grateful to Michael Temkin and Antoine Ducros for answering various questions about Berkovich spaces and to Andreas Blass for help with Remark 2.6 and Example 2.7. Conversations with Dale Cutkosky, William Gignac, Olivier Piltant and Matteo Ruggiero have also been very helpful, as have comments by Yûsuke Okuyama. Vladimir Berkovich of course deserves a special acknowledgment as neither these notes nor the summer school itself would have been possible without his work. Finally I am grateful to the organizers and the sponsors of the summer school. My research has been partially funded by grants DMS-0449465 and DMS-1001740 from the NSF.

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Jonsson, M. (2015). Dynamics on Berkovich Spaces in Low Dimensions. In: Ducros, A., Favre, C., Nicaise, J. (eds) Berkovich Spaces and Applications. Lecture Notes in Mathematics, vol 2119. Springer, Cham. https://doi.org/10.1007/978-3-319-11029-5_6

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