Skip to main content Accessibility help
×
Home

Tropical geometry and the motivic nearby fiber

  • Eric Katz (a1) and Alan Stapledon (a2)

Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Tropical geometry and the motivic nearby fiber
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Tropical geometry and the motivic nearby fiber
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Tropical geometry and the motivic nearby fiber
      Available formats
      ×

Copyright

References

Hide All
[All09]Allermann, L., Tropical intersection products on smooth varieties, arXiv:0904.2693.
[AK06]Ardila, F. and Klivans, C., The Bergman complex of a matroid and phylogenetic trees, J. Combin. Theory Ser. B 96 (2006), 3849.
[BB96]Batyrev, V. and Borisov, L., Mirror duality and string-theoretic Hodge numbers, Invent. Math. 126 (1996), 183203.
[BD96]Batyrev, V. and Dais, D., Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry, Topology 35 (1996), 901929.
[BR07]Beck, M. and Robins, S., Computing the continuous discretely (Springer, New York, 2007).
[Bit05]Bittner, F., On motivic zeta functions and the motivic nearby fiber, Math. Z. 249 (2005), 6383.
[BM03]Borisov, L. and Mavlyutov, A., String cohomology of Calabi–Yau hypersurfaces via mirror symmetry, Adv. Math. 180 (2003), 355390.
[BGS10]Burgos Gil, J. and Sombra, M., When do the recession cones of a polyhedral complex form a fan?, arXiv:1008.2608.
[Cle77]Clemens, H., Degeneration of Kähler manifolds, Duke Math. J. 44 (1977), 215290.
[DK86]Danilov, V. and Khovanskiĭ, A., Newton polyhedra and an algorithm for calculating Hodge–Deligne numbers, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 925945.
[Del71]Deligne, P., Théorie de Hodge. I, Actes du Congrès International des Mathématiciens (Nice, 1970) (Gauthier-Villars, Paris, 1971).
[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), 327348.
[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.
[GS07]Gross, M. and Siebert, B., Mirror symmetry via logarithmic degeneration data II, arxiv:0709.2290.
[Hac08]Hacking, P., The homology of tropical varieties, Collect. Math. 59 (2008), 263273.
[HK08]Helm, D. and Katz, E., Monodromy filtrations and the topology of tropical varieties, arXiv:0804.3651.
[KP09]Katz, E. and Payne, S., Realization spaces for tropical fans, in The Abel Symposium, Voss, Norway, 2009, to appear.
[Kho77]Khovanskiĭ, A., Newton polyhedra, and toroidal varieties, Funktsional. Anal. i Prilozhen. 11 (1977), 5664.
[Lan73]Landman, A., On the Picard–Lefschetz transformation for algebraic manifolds acquiring general singularities, Trans. Amer. Math. Soc. 181 (1973), 89126.
[LQ09]Luxton, M. and Qu, Z., On tropical compactifications, arXiv:0902.2009v2.
[Mik]Mikhalkin, G., Tropical geometry Texas RTG lectures, http://www.ma.utexas.edu/users/plowrey/dev/rtg/notes/.
[Mik05]Mikhalkin, G., Enumerative tropical algebraic geometry in ℝ2, J. Amer. Math. Soc. 18 (2005), 313377.
[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.
[Mor84]Morrison, D., The Clemens–Schmid exact sequence and applications, Annals of Mathematics Studies, vol. 106 (Princeton University Press, Princeton, NJ, 1984).
[MK71]Morrow, J. and Kodaira, K., Complex manifolds (Holt, New York, 1971).
[NS06]Nishinou, T. and Siebert, B., Toric degenerations of toric varieties and tropical curves, Duke Math. J. 135 (2006), 151.
[OS80]Orlik, P. and Solomon, L., Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), 167189.
[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).
[PS08]Peters, C. and Steenbrink, J., Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 52 (Springer, Berlin, 2008).
[Qu08]Qu, Z., Toric schemes over a discrete valuation ring and tropical compactifications, PhD thesis, University of Texas (2008).
[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), 289317.
[Rud09]Ruddat, H., Log Hodge groups on a toric Calabi–Yau degeneration, arXiv:0906.4809.
[Sch73]Schmid, W., Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211319.
[Spe07]Speyer, D., Uniformizing tropical curves I: genus zero and one, arXiv:0711.2677.
[Spe05]Speyer, D., Tropical geometry, PhD thesis, University of California, Berkeley (2005).
[Sta87]Stanley, R., Generalized H-vectors, intersection cohomology of toric varieties, and related results, Adv. Stud. Pure Math. 11 (1987), 187213.
[Sta97]Stanley, R., Enumerative combinatorics 1 (Cambridge University Press, Cambridge, 1997).
[Ste75/76]Steenbrink, J., Limits of Hodge structures, Invent. Math. 31 (1975/76), 229257.
[Tev07]Tevelev, J., Compactifications of subvarieties of tori, Amer. J. Math. 129 (2007), 10041087.
[Zol06]Zoładek, H., The monodromy group, in Mathematics institute of the Polish academy of sciences, Mathematical Monographs (New Series), vol. 67 (Birkhäuser, Basel, 2006).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed