  The aim of this article is to prove, using complex Abel-Jacobi maps, that the
subgroup generated by Heegner cycles associated with a fixed imaginary
quadratic field in the Griffiths group of a Kuga-Sato variety over a modular
curve has infinite rank. This generalises a classical result of Chad Schoen for
the Kuga-Sato threefold, and complements work of Amnon Besser on complex
multiplication cycles over Shimura curves. The proof relies on a formula for
the images of Heegner cycles under the complex Abel-Jacobi map given in terms
of explicit line integrals of even weight cusp forms on the complex upper
half-plane. The latter is deduced from previous joint work of the author with
Massimo Bertolini, Henri Darmon, and Kartik Prasanna by exploiting connections
with generalised Heegner cycles. As a corollary, it is proved that the
Griffiths group of the product of a Kuga-Sato variety with powers of an
elliptic curve with complex multiplication has infinite rank. This recovers
results of Ashay Burungale by a different and more direct approach.
