Abstract
We study conditions under which graph hypersurfaces admit algebraic torus operations. This leads in principle to a computation of graph motives using the theorem of Bialynicki-Birula, provided one knows the fixed point loci in a resolution of singularities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Beilinson, J. Bernstein, P. Deligne, Faisceaux perverse, in: Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque 101 Soc. Math. France, Paris, 1982, pp. 5–171.
P. Belkale, P. Brosnan, Matroids, motives, and a conjecture of Kontsevich, Duke Math. Journal 116 (2003), 147–188.
A. Bialynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. 98 (1973), 480–497.
S. Bloch, Takagi Lectures (available on Bloch's homepage) (2006).
S. Bloch, H. Esnault, D. Kreimer, On motives associated to graph polynomials, Comm. Math. Phys. 267 (2006), 181–225.
D. Broadhurst, D. Kreimer, Knots and numbers in ϕ 4 theory to 7 loops and beyond, Internat. J. Modern Phys. C6 (1995), 519–524.
D. Broadhurst, D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, Phys. Lett. B393 (1997), 403–412.
P. Brosnan, On motivic decompositions arising from the method of Bialynicki-Birula, Inventiones Math. 161 (2005), 91–111.
F. Brown, On the periods of some Feynman integrals, arXiv:0910.0114 (2009).
F. Brown, O. Schnetz, A K3 in Φ 4, Duke Math. J. 161 (2012), 1817–1862.
P. Deligne, A. Goncharov, Groupes fondamentaux motiviques de Tate mixte, Annales Sci. de l'_Ecole Norm. Sup. 38 (2005), 1–56.
R. Diestel, Graph Theory, Graduate Texts in Mathematics, Vol. 173. Springer, Heidelberg, 2012.
D. Doryn, On one example and one counterexample in counting rational points on graph hypersurfaces, Lett. Math. Phys. 97 (2011), 303–315.
F. Guillén, V. Navarro Aznar, P. Pascual-Gainza, F. Puerta, Hyperréesolutions cubiques et descente cohomologique, Lecture Notes in Mathematics, Vol. 1335, Springer-Verlag, 1988.
N. A. Karpenko, Когомологии относительных клеточных пространств и изотропных многообразий флагов, Алгебра и анализ 12 (2000), no. 1, 3–69. English transl.: N. A. Karpenko, Cohomology of relative cellular spaces and of isotropic flag varieties, St. Petersburg Math. J. 12 (2001), no. 1, 1–50
M. Kontsevich, Gelfand Seminar talk, Rutgers University (1997).
M. Levine, Tate motives and the vanishing conjectures for algebraic K-theory, in: Algebraic K-theory and Algebraic Topology (Lake Louise, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 407, Kluwer Acad. Publ., Dordrecht, 1993, pp. 167–188.
M. Levine, Mixed Motives, Mathematical Surveys and Monographs, Vol. 57, American Mathematical Society, Providence, RI, 1998.
C. Peters, J. Steenbrink, Mixed Hodge Structures, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 52, Springer-Verlag, Berlin, 2008.
J. Stembridge, Counting points on varieties over finite fields related to a conjecture of Kontsevich, Annals of Combinatorics 2 (1998), 365–385.
V. Voevodsky, E. Friedlander, A. Suslin, Cycles, Transfers, and Motivic Homology Theories, Annals of Mathematics Studies, Vol. 143, Princeton University Press, Princeton, NJ, 2000.
J. Wlodarczyk, Simple Hironaka resolution in characteristic 0, J. Amer. Math. Soc. 18 (2005), 779–822.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
MÜLLER-STACH, S., WESTRICH, B. MOTIVES OF GRAPH HYPERSURFACES WITH TORUS OPERATIONS. Transformation Groups 20, 167–182 (2015). https://doi.org/10.1007/s00031-014-9291-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-014-9291-8