Dual of a Complex 2-Torus

Question

Is a complex torus $A$ of dimension 2 always isomorphic to its dual torus (i.e. the torus obtained by taking the dual lattice), or are there counterexamples to this?

Answer 1

For non-algebraic tori, $T$ and $T^*$ are (usually) not isomorphic; for algebraic ones, they are isogeneous, and for the principally polarized abelian varieties, $T$ and $T^*$ are isomorphic.

This is obvious if you realize that a complex torus is a totally imaginary 2-dimensional complex subspace in ${\Bbb C}^4$ up to $GL(4,Z)$-action, and dual torus is a dual subspace in dual ${\Bbb C}^4$.