Math @ Duke

Publications [#345771] of Joseph D Rabinoff
Papers Published
 Baker, M; Rabinoff, J, The skeleton of the jacobian, the jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves,
International Mathematics Research Notices, vol. 2015 no. 16
(January, 2015),
pp. 74367472 [doi]
(last updated on 2021/08/04)
Abstract: Let K be an algebraically closed field which is complete with respect to a nontrivial, nonArchimedean valuation and let be its value group. Given a smooth, proper, connected Kcurve X and a skeleton of the Berkovich analytification Xan, there are two natural real tori which one can consider: the tropical Jacobian Jac() 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 AbelJacobi map is a tropical AbelJacobi map. As a consequence of these results, we deduce thatrational principal divisors on, 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 arational principal divisor on 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: R is the restriction to of.log  f  for some nonzero meromorphic function f on X if and only if F is arational tropical meromorphic function, and we use this fact to prove that there is a rational map f: X → P3 whose tropicalization, when restricted to, is an isometry onto its image.


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