@Nulhomologous : Thank you very much for your answer. Could you explain how we should interpret or change "to degenerate" here? I think somehow a half of the lattice should vanish and we consider the other half as a real lattice. Is this true?

@Nulhomologous Thank you very much. Could you more illustrate about this. In particular what is $q$ here? and why does this idea only works for elliptic curves i.e., $g=1$?

Thanks a lot. Actually I was mainly interested in the case where the complexification is a CM abelian variety and wanted to know if in this the complexification of $U$ would also be CM. But I think your example also excludes this case, so it completely answers my question.

@ Phil Tosteson. Thank you very much for your nice answer. I have another question: How about the sum of two co-edges? When do we have $e^{\vee}_1+e^{\vee}_2=0$ in $H^1(G)$? Thank you very much!

@Sam Hopkins. I don't know. What do you mean by a bridge?