The mixed Hodge structure on the fundamental group of hyperelliptic curves and higher cycles
Author:
Elisabetta Colombo
Journal:
J. Algebraic Geom. 11 (2002), 761-790
DOI:
https://doi.org/10.1090/S1056-3911-02-00332-6
Published electronically:
June 10, 2002
MathSciNet review:
1910986
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Additional Information
Abstract: In this paper we give a geometrical interpretation of an extension of mixed Hodge structures (MHS) obtained from the canonical MHS on the group ring of the fundamental group of a hyperelliptic curve modulo the fourth power of its augmentation ideal. We show that the class of this extension coincides with the regulator image of a canonical higher cycle in a hyperelliptic Jacobian. This higher cycle was introduced and studied by Collino.
[1]Be A.A.Beilinson, Higher regulators and values of L-functions, Jour. Sov. Math. 30 (1985), 2036-2070
[2]Bl S.Bloch, Algebraic cycles and higher K-theory, Adv. in Math. 61 (1986), 267-304
[3]BH J.Birman, H.Hilden, On isotopies of homeomorphisms of Riemann surfaces, Ann. of Math. 97 (1973), 424-439
[4]Ca J.A.Carlson, Extensions of mixed Hodge structures, Journeés de Geometrie Algebrique d’Angers, Sijthoff and Nordhoff, Alphen aan den Rijn (1980), 107-128
[5]Col A.Collino, Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J.Algebraic Geom. 6 (1997), 393-415
[6]FK H.M.Farkas, I.Kra, Riemann surfaces, Graduate Texts in Mathematics, 71, Springer-Verlag (1992)
[7]F W.Fulton, Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Springer-Verlag (1984)
[8]Al F.Guillén, V.Navarro Aznar, P.Pascula-Gainza, F.Puerta, Hyperreśolutions Cubiques et Descente cohomologiques, Lecture Notes in Mathematics, 1335 Springer-Verlag (1988)
[9]HB B.Harris, Harmonic volumes, Acta Math. 150 (1983), 91-123
[10]HD1 R.Hain, The geometry of the Mixed Hodge structure on the fundamental group, Proc. of Symp. in Pure Math. 46 (1987), 247-282
[11]HD2 R.Hain, The de Rham homotopy theory of complex algebraic varieties I, K-Theory 1 (1987), 271-324
[12]HD3 R.Hain, Completions of mapping class groups and the cycle $C-C^-$, Contemp. Math. 150 (1993), 75-105
[13]HD4 R.Hain, Torelli groups and geometry of moduli spaces of curves, MSRI publications 28 (1995), 97-143
[14]HL R.Hain, E.Looijenga, Mapping class groups and moduli spaces of curves, Proc. of Symp. in Pure Math. 21 (1995), 97-142
[15]J1 D.Johnson, An abelian quotient of the mapping class group $\mathcal {J}_g$, Math. Ann. 249 (1980), 225-242
[16]J2 D.Johnson, A survey of the Torelli group, Cont. Math. 20 (1983), 165-179
[17]Ka1 R.Kaenders, The mixed Hodge structure on the fundamental group of a punctured Riemann surface, Proc. Amer. Math. Soc. 129 (2000), 1271-1281
[18]Mor J.Morgan, The algebraic topology of smooth algebraic varieties, Publ. Math. IHES 48 (1978), 137-204
[19]M2 S.Morita, On the structure of the Torelli group and the Casson invariant, Topology 30 (1991), 603-621
[20]M4 S.Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993), 699-726
[21]Muller S.Muller-Stach, Constructing indecomposable motivic cohomology classes on algebraic surfaces, J. Algebraic Geom. 6 (1997), no. 3, 513-543
[22]P M.Pulte, The fundamental group of a Riemann surface: Mixed Hodge structures and algebraic cycles, Duke Math. J. 57 (1988), 721-760
[23]Sa M.Saito, Mixed Hodge modules and admissible variations C.R.Acad.Sci. 309 I (1989), 351-356
[24]SZ J.Steenbrink, S.Zucker, Variation of mixed Hodge structure I, Invent. Math. 80 (1985), 489-542
[25]Voisin C.Voisin, Variations of Hodge structure and algebraic cycles, Proc. Int. Congress of Math., Vol. 1 (Zurich, 1994), 706-715, Birkhauser, Basel (1995)
[1]Be A.A.Beilinson, Higher regulators and values of L-functions, Jour. Sov. Math. 30 (1985), 2036-2070
[2]Bl S.Bloch, Algebraic cycles and higher K-theory, Adv. in Math. 61 (1986), 267-304
[3]BH J.Birman, H.Hilden, On isotopies of homeomorphisms of Riemann surfaces, Ann. of Math. 97 (1973), 424-439
[4]Ca J.A.Carlson, Extensions of mixed Hodge structures, Journeés de Geometrie Algebrique d’Angers, Sijthoff and Nordhoff, Alphen aan den Rijn (1980), 107-128
[5]Col A.Collino, Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J.Algebraic Geom. 6 (1997), 393-415
[6]FK H.M.Farkas, I.Kra, Riemann surfaces, Graduate Texts in Mathematics, 71, Springer-Verlag (1992)
[7]F W.Fulton, Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Springer-Verlag (1984)
[8]Al F.Guillén, V.Navarro Aznar, P.Pascula-Gainza, F.Puerta, Hyperreśolutions Cubiques et Descente cohomologiques, Lecture Notes in Mathematics, 1335 Springer-Verlag (1988)
[9]HB B.Harris, Harmonic volumes, Acta Math. 150 (1983), 91-123
[10]HD1 R.Hain, The geometry of the Mixed Hodge structure on the fundamental group, Proc. of Symp. in Pure Math. 46 (1987), 247-282
[11]HD2 R.Hain, The de Rham homotopy theory of complex algebraic varieties I, K-Theory 1 (1987), 271-324
[12]HD3 R.Hain, Completions of mapping class groups and the cycle $C-C^-$, Contemp. Math. 150 (1993), 75-105
[13]HD4 R.Hain, Torelli groups and geometry of moduli spaces of curves, MSRI publications 28 (1995), 97-143
[14]HL R.Hain, E.Looijenga, Mapping class groups and moduli spaces of curves, Proc. of Symp. in Pure Math. 21 (1995), 97-142
[15]J1 D.Johnson, An abelian quotient of the mapping class group $\mathcal {J}_g$, Math. Ann. 249 (1980), 225-242
[16]J2 D.Johnson, A survey of the Torelli group, Cont. Math. 20 (1983), 165-179
[17]Ka1 R.Kaenders, The mixed Hodge structure on the fundamental group of a punctured Riemann surface, Proc. Amer. Math. Soc. 129 (2000), 1271-1281
[18]Mor J.Morgan, The algebraic topology of smooth algebraic varieties, Publ. Math. IHES 48 (1978), 137-204
[19]M2 S.Morita, On the structure of the Torelli group and the Casson invariant, Topology 30 (1991), 603-621
[20]M4 S.Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993), 699-726
[21]Muller S.Muller-Stach, Constructing indecomposable motivic cohomology classes on algebraic surfaces, J. Algebraic Geom. 6 (1997), no. 3, 513-543
[22]P M.Pulte, The fundamental group of a Riemann surface: Mixed Hodge structures and algebraic cycles, Duke Math. J. 57 (1988), 721-760
[23]Sa M.Saito, Mixed Hodge modules and admissible variations C.R.Acad.Sci. 309 I (1989), 351-356
[24]SZ J.Steenbrink, S.Zucker, Variation of mixed Hodge structure I, Invent. Math. 80 (1985), 489-542
[25]Voisin C.Voisin, Variations of Hodge structure and algebraic cycles, Proc. Int. Congress of Math., Vol. 1 (Zurich, 1994), 706-715, Birkhauser, Basel (1995)
Additional Information
Elisabetta Colombo
Affiliation:
Dipartimento di Matematica, Universita’ di Milano, via Saldini 50, 20133 Milano, Italy
Email:
elisabetta.colombo@mat.unimi.it
Received by editor(s):
August 1, 2000
Published electronically:
June 10, 2002
Additional Notes:
The author acknowledges support from MURST and GNSAGA (CNR) Italy
