Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta
- 1.2k Downloads
Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certain combinatorial objects, called skeleta. A skeleton is a metric graph embedded in the Berkovich analytification of X. A skeleton has the natural structure of a metrized complex of curves. We prove that a finite morphism of K-curves gives rise to a finite harmonic morphism of a suitable choice of skeleta. We use this to give analytic proofs of stronger ‘skeletonized’ versions of some foundational results of Liu-Lorenzini, Coleman, and Liu on simultaneous semistable reduction of curves. We then consider the inverse problem of lifting finite harmonic morphisms of metrized complexes to morphisms of curves over K. We prove that every tamely ramified finite harmonic morphism of Λ-metrized complexes of k-curves lifts to a finite morphism of K-curves. If in addition the ramification points are marked, we obtain a complete classification of all such lifts along with their automorphisms. This generalizes and provides new analytic proofs of earlier results of Saïdi and Wewers. As an application, we discuss the relationship between harmonic morphisms of metric graphs and induced maps between component groups of Néron models, providing a negative answer to a question of Ribet motivated by number theory.
This article is the first in a series of two. The second article contains several applications of our lifting results to questions about lifting morphisms of tropical curves.
KeywordsOpen Ball Harmonic Morphism Tropical Curf Finite Morphism Admissible Cover
The authors thank Amaury Thuillier and Antoine Ducros for their help with some of the descent arguments in (5.2). We are grateful to Andrew Obus for a number of useful comments based on a careful reading of the first arXiv version of this manuscript. We are also very grateful to the referee for quickly and thoroughly checking all of the technical arguments in the paper, pointing out several errors and suggesting many improvements. MB was partially supported by NSF grant DMS-1201473. EB was partially supported by the ANR-09-BLAN-0039-01.
- 1.Abramovich, S, Caporaso, L, Payne, S: The tropicalization of the moduli space of curves. to appear in Ann. Sci. Éc. Norm. Supér., preprint available at http://arxiv.org/abs/1212.0373.
- 2.Amini, O, Baker, M: Linear series on metrized complexes of algebraic curves. to appear in Math. Ann., preprint available at http://arxiv.org/abs/1204.3508.
- 4.Anand CK: A discrete analogue of the harmonic morphism, Harmonic morphisms, harmonic maps, and related topics (Brest, 1997), Chapman & Hall/CRC Res. Notes Math. 413, 109–112 (2000).Google Scholar
- 9.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
- 11.Baker, M, Payne, S, Rabinoff, J: Nonarchimedean geometry, tropicalization, and metrics on curves (2011). Preprint available at http://arxiv.org/abs/1104.0320v2.
- 12.Baker, M, Payne, S, Rabinoff, J: On the structure of non-Archimedean analytic curves. Tropical and non-A, rchimedean geometry. Contemp. Math., vol. 605, Amer. Math. Soc., Providence, RI, 2013, p. 93–121, 3204269.Google Scholar
- 13.Baker, M, Rabinoff, J: The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves. to appear in International Math. Research Notices., preprint available at http://arxiv.org/abs/1308.3864.
- 14.Berkovich, VG: Spectral theory and analytic geometry over non-Archimedean fields. Mathematical Surveys and Monographs, vol 33, American Mathematical Society, Providence, RI, 1990.Google Scholar
- 20.Bosch, S, Güntzer, U, Remmert, R: Non-Archimedean analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 261. Springer-Verlag, Berlin (1984).Google Scholar
- 21.Chan, M: Tropical hyperelliptic curves. J. Alg. Combin. 15, 2914–2955 (2012).Google Scholar
- 23.Coleman, RF: Stable maps of curves. Documenta Math. Extra volume Kato, 217–225 (2003).Google Scholar
- 24.Cornelissen, G, Kato, F, Kool, J: A combinatorial Li-Yau inequality and rational points on curves. to appear in Math. Ann., preprint available at http://arxiv.org/abs/1211.2681.
- 25.Ducros, A: La structure des courbes analytiques. Preliminary version available at http://webusers.imj-prg.fr/~antoine.ducros/livre.html.
- 31.Kontsevich, M, Soibelman, Y: Affine structures and non-Archimedean analytic spaces. The unity of mathematics, Progr. Math., Vol. 244, Birkhäuser Boston, Boston, MA (321).Google Scholar
- 33.Liu, Q: Reduction and lifting of finite covers of curves. In: Proceedings of the 2003. Workshop on Cryptography and Related Mathematics, Chuo University, pp. 161–180 (2003).Google Scholar
- 35.Mikhalkin, G: Tropical geometry and its applications. In: International Congress of Mathematicians. Vol. II, Eur. Math. Soc, pp. 827–852, Zürich (2006).Google Scholar
- 37.Mikhalkin, G, Zharkov, I: Tropical curves, their Jacobians and theta functions. Curves and abelian varieties, Contemp. Math, Vol. 465, Amer. Math. Soc., Providence, RI (2008).Google Scholar
- 40.Grothendieck, A, Raynaud, M: Revêtements étales et groupe fondamental (SGA 1), Séminaire de Géométrie Algébrique de l’Institut des Hautes Études Scientifiques (SGA 1), Société Mathématique de France, Paris (2003).Google Scholar
- 43.Temkin, M: Introduction to Berkovich analytic spaces. Preprint available at http://arxiv.org/abs/1010.2235v1.
- 44.Thuillier, A: Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d’Arakelov. Ph.D. thesis, University of Rennes (2005). Preprint available at http://tel.archives-ouvertes.fr/docs/00/04/87/50/PDF/tel-00010990.pdf.
- 46.Welliaveetil, J: A Riemann-Hurwitz formula for skeleta in non-Archimedean geometry (2013). Preprint available at http://arxiv.org/abs/1303.0164.
- 47.Wewers, S: Deformation of tame admissible covers of curves. Aspects of Galois theory (Gainesville, FL, 1996), London Math. Soc. Lecture Note Ser, Vol. 256. Cambridge Univ. Press, Cambridge (1999).Google Scholar
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.