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Forms and Currents on the Analytification of an Algebraic Variety (After Chambert-Loir and Ducros)

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Nonarchimedean and Tropical Geometry

Part of the book series: Simons Symposia ((SISY))

Abstract

Chambert-Loir and Ducros have recently introduced real differential forms and currents on Berkovich spaces. In these notes, we survey this new theory and we will compare it with tropical algebraic geometry.

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References

  1. Allermann, L., Rau, J.: First steps in tropical intersection theory. Math. Z. 264(3), 633–670 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baker, M., Rumely, R.: Potential Theory and Dynamics on the Berkovich Projective Line. Mathematical Surveys and Monographs, vol. 159. American Mathematical Society, Providence, RI (2010)

    Google Scholar 

  3. Baker, M., Payne, S., Rabinoff, J.: Nonarchimedean geometry, tropicalization, and metrics on curves. arXiv:1104.0320v2

    Google Scholar 

  4. Berkovich, V.G.: Spectral Theory and Analytic Geometry over Non-Archimedean Fields. Mathematical Surveys and Monographs, vol. 33. American Mathematical Society, Providence, RI (1990)

    Google Scholar 

  5. Berkovich, V.G.: Étale cohomology for non-Archimedean analytic spaces. Publ. Math. IHES 78, 5–161 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bosch, S., Lütkebohmert, W.: Formal and rigid geometry. I: rigid spaces. Math. Ann. 295(2), 291–317 (1993)

    MathSciNet  MATH  Google Scholar 

  7. Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry. Grundlehren der Mathematischen Wissenschaften, vol. 261. Springer, Berlin (1984)

    Google Scholar 

  8. Bourbaki, N.: Éléments de mathématique: Topologie générale. Chapitres 1 à 4. Hermann, Paris (1971)

    Google Scholar 

  9. Burgos Gil, J.I., Philippon, P., Sombra, M.: Arithmetic geometry of toric varieties. Metrics, measures and heights. Astérisque 360 (2014)

    Google Scholar 

  10. Chambert-Loir, A.: Mesure et équidistribution sur les espaces de Berkovich. J. Reine Angew. Math. 595, 215–235 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Chambert-Loir, A., Ducros, A.: Formes différentielles réelles et courants sur les espaces Berkovich. arXiv:1204.64591

    Google Scholar 

  12. Chinburg, T., Rumely, R.: The capacity pairing. J. Reine Angew. Math. 434, 1–44 (1993)

    MathSciNet  MATH  Google Scholar 

  13. Conrad, B.: Irreducible components of rigid spaces. Ann. Inst. Fourier (Grenoble) 49(2), 473–541 (1999)

    Google Scholar 

  14. Demailly, J.-P.: Complex Analytic and Differential Geometry. Version from 21 June 2012, available at the author’s homepage

    Google Scholar 

  15. Ducros, A.: Espaces de Berkovich, polytopes, squelettes et théorie des modèles. arXiv:1203.6498

    Google Scholar 

  16. Gillet, H., Soulé, C.: Arithmetic intersection theory. Publ. Math. IHES 72, 94–174 (1990)

    Article  MATH  Google Scholar 

  17. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Pure and Applied Mathematics. Wiley, New York (1978)

    MATH  Google Scholar 

  18. Gubler, W.: Local heights of subvarieties over non-Archimedean fields. J. Reine Angew. Math. 498, 61–113 (1998)

    MathSciNet  MATH  Google Scholar 

  19. Gubler, W.: A guide to tropicalizations. In: Algebraic and Combinatorial Aspects of Tropical Geometry. Contemporary Mathematics, pp. 125–189, vol. 589. American Mathematical Society, Providence, RI (2013)

    Google Scholar 

  20. Lagerberg, A.: Super currents and tropical geometry. Math. Z. 270, 1011–1050 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Samuel, P.: À propos du théorème des unités. Bull. Sci. Math. 90(2), 89–96 (1966)

    MathSciNet  MATH  Google Scholar 

  22. Sturmfels, B., Tevelev, J.: Elimination theory for tropical varieties. Math. Res. Lett. 15(2–3), 543–562 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Temkin, M.: Introduction to Berkovich analytic spaces. arXiv:1010.2235

    Google Scholar 

  24. Thuillier, A.: Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d’Arakelov. Thése de l’Université de Rennes 1 (2005)

    Google Scholar 

  25. Warner, F.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, vol. 94. Springer, New York (1983)

    Google Scholar 

  26. Zhang, S.: Admissible pairing on a curve. Invent. Math. 112, 171–193 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Universität Regensburg, Fakultät für Mathematik, Universitätsstrasse 31, 93040, Regensburg, Germany

    Walter Gubler

Authors
  1. Walter Gubler
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Correspondence to Walter Gubler .

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Editors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, USA

    Matthew Baker

  2. Department of Mathematics, Yale University, New Haven, Connecticut, USA

    Sam Payne

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Gubler, W. (2016). Forms and Currents on the Analytification of an Algebraic Variety (After Chambert-Loir and Ducros). In: Baker, M., Payne, S. (eds) Nonarchimedean and Tropical Geometry. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-319-30945-3_1

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