# Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta

@article{Amini2013LiftingHM, title={Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta}, author={Omid Amini and Matthew Baker and Erwan Brugall{\'e} and Joseph Rabinoff}, journal={Research in the Mathematical Sciences}, year={2013}, volume={2}, pages={1-67} }

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…

## 103 Citations

### Lifting harmonic morphisms II: tropical curves and metrized complexes

- Mathematics
- 2014

In this paper 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…

### Decompositions of tame profinite fundamental groups of non-archimedean curves using metrized complexes

- Mathematics
- 2018

In this paper, we study a natural covering functor from the category of tame etale coverings of a punctured curve over a complete algebraically closed non-archimedean field to the category of finite…

### A study of skeleta in non-Archimedean geometry

- Mathematics
- 2015

This thesis is a reflection of the interaction between Berkovich geometry and model theory. Using the results of Hrushovski and Loeser, we show that several interesting topological phenomena that…

### A combinatorial Li–Yau inequality and rational points on curves

- Mathematics
- 2012

We present a method to control gonality of nonarchimedean curves based on graph theory. Let $$k$$k denote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a…

### Tropical decompositions of the fundamental group of a non-archimedean curve.

- Mathematics
- 2020

In this paper we study tropicalizations of residually tame coverings of a punctured curve over a non-archimedean field. For every skeleton $\Sigma$ of a punctured curve $(X,D)$, we construct a…

### Triply mixed coverings of arbitrary base curves: Quasimodularity, quantum curves and a mysterious topological recursions

- Mathematics
- 2019

Simple Hurwitz numbers enumerate branched morphisms between Riemann surfaces with fixed ramification data. In recent years, several variants of this notion for genus $0$ base curves have appeared in…

### A generalization of the Newton-Puiseux algorithm for semistable models

- Mathematics
- 2020

In this paper we give an algorithm that calculates a skeleton of a tame covering of curves over a complete discretely valued field. The algorithm mainly relies on the {tame simultaneous semistable…

### The Skeleton of the Jacobian, the Jacobian of the Skeleton, and Lifting Meromorphic Functions From Tropical to Algebraic Curves

- Mathematics
- 2013

Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let \Lambda be its value group. Given a smooth, proper, connected K-curve X and a…

### Brill-Noether theory for curves of a fixed gonality

- MathematicsForum of Mathematics, Pi
- 2021

Abstract We prove a generalisation of the Brill-Noether theorem for the variety of special divisors
$W^r_d(C)$
on a general curve C of prescribed gonality. Our main theorem gives a closed formula…

### Riemann–Hurwitz formula for finite morphisms of p-adic curves

- Mathematics
- 2016

Given a finite morphism $$\varphi :Y\rightarrow X$$φ:Y→X of quasi-smooth Berkovich curves over a complete, non-archimedean, nontrivially valued algebraically closed field k of characteristic 0, we…

## References

SHOWING 1-10 OF 62 REFERENCES

### Lifting harmonic morphisms II: tropical curves and metrized complexes

- Mathematics
- 2014

In this paper 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…

### On the structure of nonarchimedean analytic curves

- Mathematics
- 2014

Let K be an algebraically closed, complete nonarchimedean field and let X be a smooth K-curve. In this paper we elaborate on several aspects of the structure of the Berkovich analytic space X^an. We…

### A combinatorial Li–Yau inequality and rational points on curves

- Mathematics
- 2012

We present a method to control gonality of nonarchimedean curves based on graph theory. Let $$k$$k denote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a…

### The Skeleton of the Jacobian, the Jacobian of the Skeleton, and Lifting Meromorphic Functions From Tropical to Algebraic Curves

- Mathematics
- 2013

Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let \Lambda be its value group. Given a smooth, proper, connected K-curve X and a…

### Harmonic morphisms and hyperelliptic graphs

- Mathematics
- 2007

We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula,…

### Models of Curves and Finite Covers

- MathematicsCompositio Mathematica
- 1999

Let K be a discrete valuation field with ring of integers O K .Letf : X ! Y be a finite morphism of curves over K. In this article, we study some possible relationships between the models over O K of…

### Stable reduction of finite covers of curves

- MathematicsCompositio Mathematica
- 2006

Let K be the function field of a connected regular scheme S of dimension 1, and let $f : X\to Y$ be a finite cover of projective smooth and geometrically connected curves over K with $g(X)\ge 2$.…

### Tropical hyperelliptic curves

- Mathematics
- 2011

We study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g. We define a harmonic morphism of metric graphs and prove that a metric graph is…