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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Abstract |
References |
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