(An official version is available online.)

This paper is essentially my thesis. I use a theorem of Deligne that gives a down-to-earth description of the category of ordinary abelian varieties over a finite field to give a criterion for deciding whether or not a given isogeny class of ordinary abelian varieties contains a principally polarized variety. At the end of the paper I give what I believe to be the first examples of isogeny classes of abelian varieties of dimension greater than 3 that do not contain principally polarized varieties.