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Journal Article

Cycle relations on Jacobian varieties. With an appendix by Don Zagier.


van der Geer,  Gerard
Max Planck Institute for Mathematics, Max Planck Society;

Kouvidakis,  Alexis
Max Planck Society;

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van der Geer, G., & Kouvidakis, A. (2007). Cycle relations on Jacobian varieties. With an appendix by Don Zagier. Compositio Mathematica, 143(4), 900-908.

Cite as: https://hdl.handle.net/21.11116/0000-0004-0092-6
Let X be an abelian variety over an algebraically closed field. \it A. Beauville showed [Math. Ann. 273, 647--651 (1986; Zbl 0566.14003)] that the Chow ring \textCH\Bbb Q(X) of X with rational coefficients has a double grading \textCH\Bbb Q(X)=\bigoplus \textCH^i(j)(X), where \textCH^i(j)(X)=x\in \textCH^i(X);k^*(x)= k^2i-jx for any k\in \bbfZ. Furthermore the quotient A(X) modulo algebraic equivalence inherits the double grading: A(X)=\bigoplus A^i(j)(X). Accordingly, when X is the Jacobian variety of a curve C of genus g, the class [C]\in A^g-1(X) of the image of the Abel-Jacobi mapping is decomposed as [C]=\sum^g-1j=0C(j) with C(j)\in A(j)^g-1(X). \par \it E. Colombo and \it B. van Geemen [Compos. Math. 88, No. 3, 333--353 (1993; Zbl 0802.14002)], proved that for a d-gonal curve the components C(j) vanish for j≥ d-1. Moreover, \it F. Herbaut [Compos. Math. 143, No. 4, 883--899 (2007; Zbl 1187.14006)] extended this and found cycle relations for curves having a g^rd. The main result of this paper gives simpler relations than Herbaut's and shows that \suma1+⋅s+ ar=N(a1+1)!\dots(ar+1)! C(a1)*⋅s*C(ar)=0 for N≥ d-2r+1, where ``*" denotes the Pontryagin product. In an appendix by Zagier, their relations are shown to be equivalent.