Abstract
We study the tropical Dolbeault cohomology of non-archimedean curves as defined by Chambert-Loir and Ducros. We give a precise condition for when this cohomology satisfies Poincaré duality. The condition is always satisfied when the residue field of the non-archimedean base field is the algebraic closure of a finite field. We also show that for curves over fields with residue field ℂ, the tropical (1, 1)-Dolbeault cohomology can be infinite dimensional. Our main new ingredient is an exponential type sequence that relates tropical Dolbeault cohomology to the cohomology of the sheaf of harmonic functions. As an application of our Poincaré duality result, we calculate the dimensions of the tropical Dolbeault cohomology, called tropical Hodge numbers, for (open subsets of) curves.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. G. Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Mathematical Surveys and Monographs, Vol. 33, American Mathematical Society, Providence, RI, 1990.
M. Baker, S. Payne and J. Rabinoff, On the structure of non-Archimedean analytic curves, in Tropical and Non-Archimedean Geometry, Contemporary Mathematics, Vol. 605, American Mathematical Society, Providence, RI, 2013, pp. 93–121.
R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Vol. 82, Springer, New York–Berlin, 1982.
A. Chambert-Loir and A. Ducros, Formes différentielles réelles et courants sur les espaces de Berkovich, https://doi.org/arxiv.org/abs/1204.6277.
W. Gubler, Forms and currents on the analytification of an algebraic variety (after Chambert-Loir and Ducros), in Nonarchimedean and Tropical Geometry, Simons Symposia, Springer, Cham, 2016, pp. 1–30.
P. Jell, Differential forms on Berkovich analytic spaces and their cohomology, PhD Thesis, availible at https://doi.org/epub.uni-regensburg.de/34788/1/ThesisJell.pdf.
P. Jell, A Poincaré lemma for real-valued differential forms on Berkovich spaces, Mathematische Zeitschrift 282(2016), 1149–1167.
P. Jell, K. Shaw and J. Smacka, Superforms, tropical cohomology and Poincaré duality, Advances in Geometry, to appear, https://doi.org/arxiv.org/abs/1512.07409.
P. Jell and V. Wanner, Poincaré duality for the tropical Dolbeault cohomology of non-archimedean Mumford curves, Journal of Number Theory 187 (2018), 344–371.
Y. Liu, Monodromy map for tropical Dolbeault cohomology, Algebraic Geometry, to appear, https://doi.org/users.math.yale.edu/~yl2269/monodromy.pdf.
Y. Liu, Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves, Annales Scientifiques de l’école Normale Supérieure, to appear, https://doi.org/users.math.yale.edu/~yl2269/deRham.pdf.
G. Mikhalkin and I. Zharkov, Tropical curves, their Jacobians and theta functions, in Curves and Abelian Varieties, Contemporary Mathematics, Vol. 465, American Mathematical Society, Providence, RI, 2008, pp. 203–230.
A. Thuillier, Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la théorie d’Arakelov, https://doi.org/tel.archives-ouvertes.fr/file/index/docid/48750/filename/tel-00010990.pdf.
V. Wanner, Harmonic functions on the Berkovich projective line, Master Thesis, https://doi.org/epub.uni-regensburg.de/35758/1/MA.pdf.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was supported by the collaborative research center CRC 1085 “Higher Invariants” funded by the Deutsche Forschungsgemeinschaft.
Rights and permissions
About this article
Cite this article
Jell, P. Tropical Hodge numbers of non-archimedean curves. Isr. J. Math. 229, 287–305 (2019). https://doi.org/10.1007/s11856-018-1799-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-018-1799-5