  The cones of divisors and curves defined by various positivity conditions on
a smooth projective variety have been the subject of a great deal of work in
algebraic geometry, and by now they are quite well understood. However the
analogous cones for cycles of higher codimension and dimension have started to
come into focus only recently. The purpose of this paper is to explore some of
the phenomena that can occur by working out the picture fairly completely in a
couple of simple but non-trivial cases. Specifically, we study cycles of
arbitrary codimension on the self-product of an elliptic curve with complex
multiplication, as well as two dimensional cycles on the product of a very
general abelian surface with itself. Already one finds various non-classical
behavior, for instance nef cycles that fail to be pseudoeffective: this answers
a question raised in 1964 by Grothendieck in correspondence with Mumford. We
also propose a substantial number of open problems for further investigation.
