26 research outputs found

### The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves

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 skeleton \Gamma of the Berkovich
analytification X^\an, there are two natural real tori which one can consider:
the tropical Jacobian Jac(\Gamma) and the skeleton of the Berkovich
analytification Jac(X)^\an. We show that the skeleton of the Jacobian is
canonically isomorphic to the Jacobian of the skeleton as principally polarized
tropical abelian varieties. In addition, we show that the tropicalization of a
classical Abel-Jacobi map is a tropical Abel-Jacobi map. As a consequence of
these results, we deduce that \Lambda-rational principal divisors on \Gamma, in
the sense of tropical geometry, are exactly the retractions of principal
divisors on X. We actually prove a more precise result which says that,
although zeros and poles of divisors can cancel under the retraction map, in
order to lift a \Lambda-rational principal divisor on \Gamma to a principal
divisor on X it is never necessary to add more than g extra zeros and g extra
poles. Our results imply that a continuous function F:\Gamma -> R is the
restriction to \Gamma of -log|f| for some nonzero meromorphic function f on X
if and only if F is a \Lambda-rational tropical meromorphic function, and we
use this fact to prove that there is a rational map f : X --> P^3 whose
tropicalization, when restricted to \Gamma, is an isometry onto its image.Comment: 21 pages, 1 figur

### Skeletons and tropicalizations

Let $K$ be a complete, algebraically closed non-archimedean field with ring
of integers $K^\circ$ and let $X$ be a $K$-variety. We associate to the data of
a strictly semistable $K^\circ$-model $\mathscr X$ of $X$ plus a suitable
horizontal divisor $H$ a skeleton $S(\mathscr X,H)$ in the analytification of
$X$. This generalizes Berkovich's original construction by admitting unbounded
faces in the directions of the components of H. It also generalizes
constructions by Tyomkin and Baker--Payne--Rabinoff from curves to higher
dimensions. Every such skeleton has an integral polyhedral structure. We show
that the valuation of a non-zero rational function is piecewise linear on
$S(\mathscr X, H)$. For such functions we define slopes along codimension one
faces and prove a slope formula expressing a balancing condition on the
skeleton. Moreover, we obtain a multiplicity formula for skeletons and
tropicalizations in the spirit of a well-known result by Sturmfels--Tevelev. We
show a faithful tropicalization result saying roughly that every skeleton can
be seen in a suitable tropicalization. We also prove a general result about
existence and uniqueness of a continuous section to the tropicalization map on
the locus of tropical multiplicity one.Comment: 44 pages, 2 figures. Version 3: minor errors corrected; Remark 3.14
expanded. Final version, to appear in Advances in Mathematic

### On the structure of nonarchimedean analytic curves

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 define semistable vertex
sets of X^an and their associated skeleta, which are essentially finite metric
graphs embedded in X^an. We prove a folklore theorem which states that
semistable vertex sets of X are in natural bijective correspondence with
semistable models of X, thus showing that our notion of skeleton coincides with
the standard definition of Berkovich. We use the skeletal theory to define a
canonical metric on H(X^an) := X^an - X(K), and we give a proof of Thuillier's
nonarchimedean Poincar\'e-Lelong formula in this language using results of
Bosch and L\"utkebohmert.Comment: 23 pages. This an expanded version of section 5 of arXiv:1104.0320
which appears in the conference proceedings "Tropical and Non-Archimedean
Geometry