Abstract
This is an expository article on the adic tropicalization of algebraic varieties. We outline joint work with Sam Payne in which we put a topology and structure sheaf of local topological rings on the exploded tropicalization. The resulting object, which blends polyhedral data of the tropicalization with algebraic data of the associated initial degenerations, is called the adic tropicalization. It satisfies a theorem of the form “Huber analytification is the limit of all adic tropicalizations.” We explain this limit theorem in the present article, and illustrate connections between adic tropicalization and the curve complexes of O. Amini and M. Baker.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Amini, O., Baker, M.: Linear series on metrized complexes of algebraic curves. arXiv: 1204.3508 (2012)
Berkovich, V.G.: Spectral Theory and Analytic Geometry Over Non-Archimedean Fields. Mathematical Surveys and Monographs, vol. 33. American Mathematical Society, Providence, RI (1990)
Bosch, S.: Lectures on Formal and Rigid Geometry. Lecture Notes in Mathematics, vol. 2105. Springer, Cham (2014)
Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence, RI (2011)
Foster, T., Payne, S.: Limits of tropicalizations II: structure sheaves, adic spaces and cofinality of Gubler models (in preparation)
Foster, T., Gross, P., Payne, S.: Limits of tropicalizations. Isr. J. Math. 201 (2), 835–846 (2014)
Gillam, W.D.: Localization of ringed spaces. Adv. Pure Math. 1 (5), 250–263 (2011)
Gross, M.: Tropical Geometry and Mirror Symmetry. CBMS Regional Conference Series in Mathematics, vol. 114. American Mathematical Society, Providence, RI (2011); Published for the Conference Board of the Mathematical Sciences, Washington, DC. http://www.ams.org/mathscinet/search/publdoc.html (2011)
Gross, M., Siebert, B.: Affine manifolds, log structures, and mirror symmetry. Turk. J. Math. 27 (1), 33–60 (2003)
Gubler, W.: A guide to tropicalizations. In: Algebraic and Combinatorial Aspects of Tropical Geometry. Contemporary Mathematics, vol. 589, pp. 125–189. American Mathematical Society, Providence, RI (2013)
Gubler, W., Soto, A.: Classification of normal toric varieties over a valuation ring of rank one. arXiv: 1303.1987 (2013)
Hartshorne, R.: Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer, New York (1977)
Huber, R.: A generalization of formal schemes and rigid analytic varieties. Math. Z. 217 (4), 513–551 (1994)
Huber, R.: Étale Cohomology of Rigid Analytic Varieties and Adic Spaces. Aspects of Mathematics, vol. E30. Friedr. Vieweg & Sohn, Braunschweig (1996)
Kempf, G., Knudsen, F.F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings. I. Lecture Notes in Mathematics, vol. 339. Springer, Berlin (1973)
Kontsevich, M., Soibelman, Y.: Homological mirror symmetry and torus fibrations. In: Symplectic Geometry and Mirror Symmetry (Seoul, 2000), pp. 203–263. World Science Publisher, River Edge, NJ (2001)
Kontsevich, M., Soibelman, Y.: Affine structures and non-Archimedean analytic spaces. In: The Unity of Mathematics. Progress in Mathematics, vol. 244, pp. 321–385. Birkhäuser, Boston, MA (2006)
Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. Mathematical Surveys and Monographs, vol. 161. American Mathematical Society, Providence, RI (2015)
Parker, B.: Exploded fibrations. In: Proceedings of Gökova Geometry-Topology Conference 2006, Gökova Geometry/Topology Conference (GGT), Gökova, pp. 52–90 (2007)
Parker, B.: Gromov Witten invariants of exploded manifolds. arXiv: 1102.0158 (2011)
Parker, B. Exploded manifolds. Adv. Math. 229 (6), 3256–3319 (2012)
Parker, B.: Log geometry and exploded manifolds. Abh. Math. Semin. Univ. Hambg. 82 (1), 43–81 (2012)
Payne, S.: Analytification is the limit of all tropicalizations. Math. Res. Lett. 16 (3), 543–556 (2009)
Payne, S.: Fibers of tropicalization. Math. Z. 262 (2), 301–311 (2009)
Payne, S.: Erratum to: Fibers of tropicalization. Math. Z. 272 (3–4), 1403–1406 (2012)
Scholze, P.: Perfectoid spaces. Publ. Math. Inst. Hautes Études Sci. 116, 245–313 (2012)
van der Put, M., Schneider, P.: Points and topologies in rigid geometry. Math. Ann. 302 (1), 81–103 (1995)
Wedhorn, T.: Adic spaces. http://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/wedhorn/Lehre/AdicSpaces.pdf (2012)
Włodarczyk, J.: Embeddings in toric varieties and prevarieties. J. Algebraic Geom. 2 (4), 705–726 (1993)
Acknowledgements
The author would like to thank the Simons Foundation and the organizers of the Simons Symposium on non-Archimedean and Tropical Geometry for bringing together a fantastic group of researchers for an incredibly stimulating week of mathematics. Special thanks go to S. Payne, as this paper is a report on joint with him. The author thanks M. Baker, F. Baldassarri, W. Gubler, and D. Ranganathan for helpful remarks. Finally, the author thanks the reviewer for comments that greatly improved this paper.
The author is supported by NSF RTG grant DMS-0943832.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Foster, T. (2016). Introduction to Adic Tropicalization. In: Baker, M., Payne, S. (eds) Nonarchimedean and Tropical Geometry. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-319-30945-3_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-30945-3_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-30944-6
Online ISBN: 978-3-319-30945-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)