Vol. 9, No. 2, 2015

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Lifting harmonic morphisms II: Tropical curves and metrized complexes

Omid Amini, Matthew Baker, Erwan Brugallé and Joseph Rabinoff

Vol. 9 (2015), No. 2, 267–315
Abstract

We prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the nonvanishing of certain Hurwitz numbers, and we give various conditions under which this obstruction does vanish. In particular, we show that any finite harmonic morphism of (nonaugmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from C to the tropical projective line are equivalent. We study liftability of metrized complexes equipped with a finite group action, and use this to classify all augmented metric graphs arising as the tropicalization of a hyperelliptic curve. We prove that there exists a d-gonal tropical curve that does not lift to a d-gonal algebraic curve.

This article is the second in a series of two.

Keywords
tropical lifting, skeleton, Berkovich space, analytic curve, harmonic morphism, Hurwitz number, metrized complex
Mathematical Subject Classification 2010
Primary: 14G22
Secondary: 14T05, 11G20
Milestones
Received: 25 June 2013
Revised: 9 July 2014
Accepted: 6 December 2014
Published: 5 March 2015
Authors
Omid Amini
CNRS
Département de mathématiques et applications
École Normale Supérieure
45 Rue d’Ulm
75005 Paris
France
Matthew Baker
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
United States
Erwan Brugallé
Université Pierre et Marie Curie (Paris 6)
4 Place Jussieu
75005 Paris
France
Joseph Rabinoff
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
United States