Cycles positifs dans les vari\'et\'es ab\'eliennes

In the first part, we study the structure of the R-algebra generated by the Hodge classes on the self-product A^e of a very general principally polarized abelian variety A. In the second part, we compare various notions of positivity for cycles of higher codimension in A^e. In particular, we prove that, in every codimension 1 < k < en-1, there exist classes that are numerically effective but not pseudoeffective, which generalises a result of Debarre, Ein, Lazarsfeld and Voisin.