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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Algebraic cycles and the Hodge structure of a Kuga fiber variety
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by B. Brent Gordon PDF
Trans. Amer. Math. Soc. 336 (1993), 933-947 Request permission

Abstract:

Let $\tilde A$ denote a smooth compactification of the $k$-fold fiber product of the universal family ${A^1} \to M$ of elliptic curves with level $N$ structure. The purpose of this paper is to completely describe the algebraic cycles in and the Hodge structure of the Betti cohomology ${H^{\ast } }(\tilde A,\mathbb {Q})$ of $\tilde A$ , for by doing so we are able (a) to verify both the usual and generalized Hodge conjectures for $\tilde A$ ; (b) to describe both the kernel and the image of the Abel-Jacobi map from algebraic cycles algebraically equivalent to zero (modulo rational equivalence) into the Griffiths intermediate Jacobian; and (c) to verify Tate’s conjecture concerning the algebraic cycles in the étale cohomology $H_{{\text {et}}}^{\ast } (\tilde A \otimes \bar {\mathbb {Q}},{\mathbb {Q}_l})$. The methods used lead also to a complete description of the Hodge structure of the Betti cohomology ${H^{\ast } }({E^k},\mathbb {Q})$ of the $k$-fold product of an elliptic curve $E$ without complex multiplication, and a verification of the generalized Hodge conjecture for ${E^k}$ .
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 336 (1993), 933-947
  • MSC: Primary 14C30; Secondary 14C25, 14F20, 14K30
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1097167-2
  • MathSciNet review: 1097167