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The aim of this article is to prove, using complex Abel-Jacobi maps, that thesubgroup generated by Heegner cycles associated with a fixed imaginaryquadratic field in the Griffiths group of a Kuga-Sato variety over a modularcurve has infinite rank. This generalises a classical result of Chad Schoen forthe Kuga-Sato threefold, and complements work of Amnon Besser on complexmultiplication cycles over Shimura curves. The proof relies on a formula forthe images of Heegner cycles under the complex Abel-Jacobi map given in termsof explicit line integrals of even weight cusp forms on the complex upperhalf-plane. The latter is deduced from previous joint work of the author withMassimo Bertolini, Henri Darmon, and Kartik Prasanna by exploiting connectionswith generalised Heegner cycles. As a corollary, it is proved that theGriffiths group of the product of a Kuga-Sato variety with powers of anelliptic curve with complex multiplication has infinite rank. This recoversresults of Ashay Burungale by a different and more direct approach.

Heegner cycles in Griffiths groups of Kuga-Sato varieties