Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T16:42:35.546Z Has data issue: false hasContentIssue false
  • English
  • Français

Tropical geometry and the motivic nearby fiber

Part of: Families, fibrations

Published online by Cambridge University Press:  09 November 2011

Eric Katz  and
Alan Stapledon
Eric Katz
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 (email: eekatz@math.uwaterloo.ca)
Alan Stapledon
Affiliation:
Department of Mathematics, University of British Columbia, BC, Canada V6T 1Z2 (email: astapldn@gmail.com)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the ‘tropical motivic nearby fiber’. This invariant specializes in the schön case to the Hodge–Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We give purely combinatorial expressions for this Hodge–Deligne polynomial in the cases of schön hypersurfaces and matroidal tropical varieties. We also deduce a formula for the Euler characteristic of a general fiber of the degeneration.

Keywords

tropical geometrymonodromyHodge theory

MSC classification

Secondary: 14T05: Tropical geometry 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 1 , January 2012 , pp. 269 - 294
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[All09]Allermann, L., Tropical intersection products on smooth varieties, arXiv:0904.2693.Google Scholar
[AK06]Ardila, F. and Klivans, C., The Bergman complex of a matroid and phylogenetic trees, J. Combin. Theory Ser. B 96 (2006), 38–49.CrossRefGoogle Scholar
[BB96]Batyrev, V. and Borisov, L., Mirror duality and string-theoretic Hodge numbers, Invent. Math. 126 (1996), 183–203.CrossRefGoogle Scholar
[BD96]Batyrev, V. and Dais, D., Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry, Topology 35 (1996), 901–929.CrossRefGoogle Scholar
[BR07]Beck, M. and Robins, S., Computing the continuous discretely (Springer, New York, 2007).Google Scholar
[Bit05]Bittner, F., On motivic zeta functions and the motivic nearby fiber, Math. Z. 249 (2005), 63–83.CrossRefGoogle Scholar
[BM03]Borisov, L. and Mavlyutov, A., String cohomology of Calabi–Yau hypersurfaces via mirror symmetry, Adv. Math. 180 (2003), 355–390.CrossRefGoogle Scholar
[BGS10]Burgos Gil, J. and Sombra, M., When do the recession cones of a polyhedral complex form a fan?, arXiv:1008.2608.Google Scholar
[Cle77]Clemens, H., Degeneration of Kähler manifolds, Duke Math. J. 44 (1977), 215–290.CrossRefGoogle Scholar
[DK86]Danilov, V. and Khovanskiĭ, A., Newton polyhedra and an algorithm for calculating Hodge–Deligne numbers, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 925–945.Google Scholar
[Del71]Deligne, P., Théorie de Hodge. I, Actes du Congrès International des Mathématiciens (Nice, 1970) (Gauthier-Villars, Paris, 1971).Google Scholar
[DL01]Denef, J. and Loeser, F., Geometry on arc spaces of algebraic varieties, in European congress of mathematics, Vol. I, (Barcelona, 2000), Progress in Mathematics, vol. 201 (Birkhäuser, Basel, 2001), 327–348.Google Scholar
[Ful93]Fulton, W., Introduction to toric varieties, Annals of Mathematics Studies, vol. 131 (Princeton University Press, Princeton, NJ, 1993), The William H. Roever Lectures in Geometry.CrossRefGoogle Scholar
[GS07]Gross, M. and Siebert, B., Mirror symmetry via logarithmic degeneration data II, arxiv:0709.2290.Google Scholar
[Hac08]Hacking, P., The homology of tropical varieties, Collect. Math. 59 (2008), 263–273.CrossRefGoogle Scholar
[HK08]Helm, D. and Katz, E., Monodromy filtrations and the topology of tropical varieties, arXiv:0804.3651.Google Scholar
[KP09]Katz, E. and Payne, S., Realization spaces for tropical fans, in The Abel Symposium, Voss, Norway, 2009, to appear.Google Scholar
[Kho77]Khovanskiĭ, A., Newton polyhedra, and toroidal varieties, Funktsional. Anal. i Prilozhen. 11 (1977), 56–64.Google Scholar
[Lan73]Landman, A., On the Picard–Lefschetz transformation for algebraic manifolds acquiring general singularities, Trans. Amer. Math. Soc. 181 (1973), 89–126.CrossRefGoogle Scholar
[LQ09]Luxton, M. and Qu, Z., On tropical compactifications, arXiv:0902.2009v2.Google Scholar
[Mik]Mikhalkin, G., Tropical geometry Texas RTG lectures, http://www.ma.utexas.edu/users/plowrey/dev/rtg/notes/.Google Scholar
[Mik05]Mikhalkin, G., Enumerative tropical algebraic geometry in ℝ2, J. Amer. Math. Soc. 18 (2005), 313–377.CrossRefGoogle Scholar
[Mik08]Mikhalkin, G., Moduli spaces of rational tropical curves, in Proc. Gökova geometry–topology conference 2007, Gökova, 28 May–2 June 2007, eds. S. Akbulut, T. Önder and D. Auroux (International Press, Boston, MA, 2008), 39–51.Google Scholar
[Mor84]Morrison, D., The Clemens–Schmid exact sequence and applications, Annals of Mathematics Studies, vol. 106 (Princeton University Press, Princeton, NJ, 1984).Google Scholar
[MK71]Morrow, J. and Kodaira, K., Complex manifolds (Holt, New York, 1971).Google Scholar
[NS06]Nishinou, T. and Siebert, B., Toric degenerations of toric varieties and tropical curves, Duke Math. J. 135 (2006), 1–51.CrossRefGoogle Scholar
[OS80]Orlik, P. and Solomon, L., Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167–189.CrossRefGoogle Scholar
[PS07]Peters, C. and Steenbrink, J., Hodge number polynomials for nearby and vanishing cohomology, in Algebraic cycles and motives, Vol. 2, London Mathematical Society Lecture Notes Series, vol. 344 (Cambridge University Press, Cambridge, 2007).Google Scholar
[PS08]Peters, C. and Steenbrink, J., Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 52 (Springer, Berlin, 2008).Google Scholar
[Qu08]Qu, Z., Toric schemes over a discrete valuation ring and tropical compactifications, PhD thesis, University of Texas (2008).Google Scholar
[RST05]Richter-Gebert, J., Sturmfels, B. and Theobald, T., First steps in tropical geometry, in Idempotent mathematics and mathematical physics, Contemporary Mathematics, vol. 377 (American Mathematical Society, Providence, RI, 2005), 289–317.CrossRefGoogle Scholar
[Rud09]Ruddat, H., Log Hodge groups on a toric Calabi–Yau degeneration, arXiv:0906.4809.Google Scholar
[Sch73]Schmid, W., Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211–319.CrossRefGoogle Scholar
[Spe07]Speyer, D., Uniformizing tropical curves I: genus zero and one, arXiv:0711.2677.Google Scholar
[Spe05]Speyer, D., Tropical geometry, PhD thesis, University of California, Berkeley (2005).Google Scholar
[Sta87]Stanley, R., Generalized H-vectors, intersection cohomology of toric varieties, and related results, Adv. Stud. Pure Math. 11 (1987), 187–213.CrossRefGoogle Scholar
[Sta97]Stanley, R., Enumerative combinatorics 1 (Cambridge University Press, Cambridge, 1997).CrossRefGoogle Scholar
[Ste75/76]Steenbrink, J., Limits of Hodge structures, Invent. Math. 31 (1975/76), 229–257.CrossRefGoogle Scholar
[Tev07]Tevelev, J., Compactifications of subvarieties of tori, Amer. J. Math. 129 (2007), 1004–1087.CrossRefGoogle Scholar
[Zol06]Zoładek, H., The monodromy group, in Mathematics institute of the Polish academy of sciences, Mathematical Monographs (New Series), vol. 67 (Birkhäuser, Basel, 2006).Google Scholar