I've read in (abstracts of) papers that there are abelian varieties over fields of positive characteristic that admit no prinicipal polarization. Apparently its not the easiest thing to find an example of, but I was thinking it should be much easier over the complex numbers.

All abelian varieties of dimension 1 are elliptic curves which always have a principal polarization. So any example would have to be two dimensional. So my question is given an abelian variety with a polarization, is there a good way of telling if there is or isn't a principal polarization?

Or if that's in general a difficult question, are there some relatively simple examples where you can really see that there are or aren't any principal polarizations?

I've read in (abstracts of) papers that there are abelian varieties over fields of positive characteristic that admit no prinicipal polarization. Apparently its not the easiest thing to find an example of, but I was thinking it should be much easier over the complex numbers.

All abelian varieties of dimension 1 are elliptic curves which always have a principal polarization. So any example would have to be at least two dimensional. So my question is given an abelian variety with a polarization, is there a good way of telling if there is or isn't a principal polarization?

Or if that's in general a difficult question, are there some relatively simple examples where you can really see that there are or aren't any principal polarizations?