A tropical abelian variety is given by a quotient of a real vector space $V \cong \mathbb{R}^g$ with a fixed integral structure $\Gamma_2$, by a lattice $\Gamma_1$, equipped with some aditional structure (polarization). How precisely can this be seen as a limit of a family of complex abelian varieties (in terms of their lattices)?
There is such a limit sketched by the end of the first page of this paper: https://arxiv.org/pdf/2002.02347.pdf , but I cannot see what this means (if, for example, $\gamma$ generates $\Gamma_1$ and $\Gamma_2=\mathbb{Z} \subset \mathbb{R}$ than $X_\epsilon = \mathbb{C}^*/{\epsilon^{-1} e^{\gamma\mathbb{Z}}}$ and $\epsilon^{-1}e^{\gamma \mathbb{Z}}$ is not even a subgroup of $\mathbb{C}^*$ for most of $\epsilon$.) On the other hand this does not seem complicated at all so I have probably missed something.
I also read in the literature that this degeneration goes back to Mumford but his paper seems a bit involved and uses different language. I asked this question on Math Stack Exchange but maybe it is more suitable here.
