Polylogarithms, regulators, and Arakelov motivic complexes

Goncharov, A.

doi:10.1090/S0894-0347-04-00472-2

Polylogarithms, regulators, and Arakelov motivic complexes
HTML articles powered by AMS MathViewer

by A. B. Goncharov

J. Amer. Math. Soc. 18 (2005), 1-60

DOI: https://doi.org/10.1090/S0894-0347-04-00472-2

Published electronically: November 3, 2004

PDF | Request permission

Abstract:

We construct an explicit regulator map from the weight $n$ Bloch higher Chow group complex to the weight $n$ Deligne complex of a regular projective complex algebraic variety $X$. We define the weight $n$ Arakelov motivic complex as the cone of this map shifted by one. Its last cohomology group is (a version of) the Arakelov Chow group defined by H. Gillet and C. Soulé. We relate the Grassmannian $n$-logarithms to the geometry of the symmetric space $SL_n(\mathcal {C})/SU(n)$. For $n=2$ we recover Lobachevsky’s formula expressing the volume of an ideal geodesic simplex in the hyperbolic space via the dilogarithm. Using the relationship with symmetric spaces we construct the Borel regulator on $K_{2n-1}(\mathcal {C})$ via the Grassmannian $n$-logarithms. We study the Chow dilogarithm and prove a reciprocity law which strengthens Suslin’s reciprocity law for Milnor’s group $K^M_3$ on curves. Our note,“Chow polylogarithms and regulators”, can serve as an introduction to this paper.

References

V. I. Arnol′d, Matematicheskie metody klassicheskoĭ mekhaniki, Izdat. “Nauka”, Moscow, 1974 (Russian). MR 0474390
A. A. Beĭlinson, Higher regulators and values of $L$-functions, Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 181–238 (Russian). MR 760999
A. Beĭlinson, Height pairing between algebraic cycles, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 1–24. MR 902590, DOI 10.1090/conm/067/902590
A. A. Beĭlinson, Notes on absolute Hodge cohomology, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 35–68. MR 862628, DOI 10.1090/conm/055.1/862628
A. Beĭlinson and P. Deligne, Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 97–121 (French). MR 1265552, DOI 10.1090/pspum/055.2/1265552
A. Beĭlinson, R. MacPherson, and V. Schechtman, Notes on motivic cohomology, Duke Math. J. 54 (1987), no. 2, 679–710. MR 899412, DOI 10.1215/S0012-7094-87-05430-5
Spencer Bloch, Algebraic cycles and higher $K$-theory, Adv. in Math. 61 (1986), no. 3, 267–304. MR 852815, DOI 10.1016/0001-8708(86)90081-2
S. Bloch, The moving lemma for higher Chow groups, J. Algebraic Geom. 3 (1994), no. 3, 537–568. MR 1269719
Spencer Bloch, Algebraic cycles and the Beĭlinson conjectures, The Lefschetz centennial conference, Part I (Mexico City, 1984) Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1986, pp. 65–79. MR 860404, DOI 10.1090/conm/058.1/860404
Spencer Bloch and Igor Kříž, Mixed Tate motives, Ann. of Math. (2) 140 (1994), no. 3, 557–605. MR 1307897, DOI 10.2307/2118618
Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235–272 (1975). MR 387496, DOI 10.24033/asens.1269
Armand Borel, Cohomologie de $\textrm {SL}_{n}$ et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 4, 613–636 (French). MR 506168
Raoul Bott, On the characteristic classes of groups of diffeomorphisms, Enseign. Math. (2) 23 (1977), no. 3-4, 209–220. MR 488080
Jose Ignacio Burgos, Arithmetic Chow rings and Deligne-Beilinson cohomology, J. Algebraic Geom. 6 (1997), no. 2, 335–377. MR 1489119
Caterina Consani, Double complexes and Euler $L$-factors, Compositio Math. 111 (1998), no. 3, 323–358. MR 1617133, DOI 10.1023/A:1000362027455
Johan L. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Topology 15 (1976), no. 3, 233–245. MR 413122, DOI 10.1016/0040-9383(76)90038-0
P. Deligne, Le déterminant de la cohomologie, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 93–177 (French). MR 902592, DOI 10.1090/conm/067/902592
William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
E. B. Dynkin, Topological characteristics of homomorphisms of compact Lie groups, Mat. Sb. (N.S.) 35(77) (1954), 129–173 (Russian). MR 0067124
A. M. Gabrièlov, I. M. Gel′fand, and M. V. Losik, Combinatorial computation of characteristic classes. I, II, Funkcional. Anal. i Priložen. 9 (1975), no. 2, 12–28; ibid. 9 (1975), no. 3, 5–26 (Russian). MR 0410758
I. M. Gel′fand and R. D. MacPherson, Geometry in Grassmannians and a generalization of the dilogarithm, Adv. in Math. 44 (1982), no. 3, 279–312. MR 658730, DOI 10.1016/0001-8708(82)90040-8
Henri Gillet and Christophe Soulé, Arithmetic intersection theory, Inst. Hautes Études Sci. Publ. Math. 72 (1990), 93–174 (1991). MR 1087394
A. B. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995), no. 2, 197–318. MR 1348706, DOI 10.1006/aima.1995.1045
A. B. Goncharov, Polylogarithms and motivic Galois groups, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 43–96. MR 1265551, DOI 10.1090/pspum/055.2/1265551
A. B. Goncharov, Explicit construction of characteristic classes, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 169–210. MR 1237830
A. B. Goncharov, Deninger’s conjecture of $L$-functions of elliptic curves at $s=3$, J. Math. Sci. 81 (1996), no. 3, 2631–2656. Algebraic geometry, 4. MR 1420221, DOI 10.1007/BF02362333
A. B. Goncharov, Chow polylogarithms and regulators, Math. Res. Lett. 2 (1995), no. 1, 95–112. MR 1312980, DOI 10.4310/MRL.1995.v2.n1.a9
A. B. Goncharov, Geometry of the trilogarithm and the motivic Lie algebra of a field, Regulators in analysis, geometry and number theory, Progr. Math., vol. 171, Birkhäuser Boston, Boston, MA, 2000, pp. 127–165. MR 1724890
A. B. Goncharov, Explicit regulator maps on polylogarithmic motivic complexes, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998) Int. Press Lect. Ser., vol. 3, Int. Press, Somerville, MA, 2002, pp. 245–276. MR 1978709
A. B. Goncharov and J. Zhao, Grassmannian trilogarithms, Compositio Math. 127 (2001), no. 1, 83–108. MR 1832988, DOI 10.1023/A:1017504115184
Richard M. Hain and Robert MacPherson, Higher logarithms, Illinois J. Math. 34 (1990), no. 2, 392–475. MR 1046570
Richard M. Hain, The existence of higher logarithms, Compositio Math. 100 (1996), no. 3, 247–276. MR 1387666
Richard M. Hain and Jun Yang, Real Grassmann polylogarithms and Chern classes, Math. Ann. 304 (1996), no. 1, 157–201. MR 1367888, DOI 10.1007/BF01446290
Masaki Hanamura and Robert MacPherson, Geometric construction of polylogarithms, Duke Math. J. 70 (1993), no. 3, 481–516. MR 1224097, DOI 10.1215/S0012-7094-93-07010-X
Masaki Hanamura and Robert MacPherson, Geometric construction of polylogarithms. II, Functional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993) Progr. Math., vol. 132, Birkhäuser Boston, Boston, MA, 1996, pp. 215–282. MR 1389020
G. W. Leibniz, Mathematische Schriften. Bd. I.: Briefwechsel zwischen Leibniz und Oldenburg, Collins, Newton, Galloys, Vitale Giordano. Bd. II: Briefwechsel zwischen Leibniz, Hugens van Zulichem und dem Marquis de l’Hospital, Georg Olms Verlagsbuchhandlung, Hildesheim, 1962 (German). Herausgegeben von C. I. Gerhardt; Two volumes bound as one. MR 0141575

Notes on $\mathbb {R}$-Hodge-Tate sheaves

Marc Levine, Bloch’s higher Chow groups revisited, Astérisque 226 (1994), 10, 235–320. $K$-theory (Strasbourg, 1992). MR 1317122
Stephen Lichtenbaum, Values of zeta-functions, étale cohomology, and algebraic $K$-theory, Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 489–501. MR 0406981
Robert MacPherson, The combinatorial formula of Gabrielov, Gel′fand and Losik for the first Pontrjagin class, Séminaire Bourbaki, 29e année (1976/77), Lecture Notes in Math., vol. 677, Springer, Berlin, 1978, pp. Exp. No. 497, pp. 105–124. MR 521763
Jan Nekovář, Beĭlinson’s conjectures, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 537–570. MR 1265544, DOI 10.1090/pspum/055.1/1265544
Yu. P. Nesterenko and A. A. Suslin, Homology of the general linear group over a local ring, and Milnor’s $K$-theory, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 121–146 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 121–145. MR 992981, DOI 10.1070/IM1990v034n01ABEH000610
M. Rapoport, N. Schappacher, and P. Schneider (eds.), Beilinson’s conjectures on special values of $L$-functions, Perspectives in Mathematics, vol. 4, Academic Press, Inc., Boston, MA, 1988. MR 944987
Anthony J. Scholl, Integral elements in $K$-theory and products of modular curves, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 467–489. MR 1744957
C. Soulé, Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge University Press, Cambridge, 1992. With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer. MR 1208731, DOI 10.1017/CBO9780511623950
A. A. Suslin, Homology of $\textrm {GL}_{n}$, characteristic classes and Milnor $K$-theory, Algebraic $K$-theory, number theory, geometry and analysis (Bielefeld, 1982) Lecture Notes in Math., vol. 1046, Springer, Berlin, 1984, pp. 357–375. MR 750690, DOI 10.1007/BFb0072031
Don Zagier, Polylogarithms, Dedekind zeta functions and the algebraic $K$-theory of fields, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 391–430. MR 1085270, DOI 10.1007/978-1-4612-0457-2_{1}9