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There is a similar interesting question here

which has not been answered. I therefore ask this question in the hope to get an answer. I wonder how a family of complex abelian varieties can exactly degenerate to a real tropical abelian variety? A tropical abelian vareity is simply a real torus $\mathbb{R}^g/\Lambda$, where $\Lambda$ is a lattice in $\mathbb{R}^g$ of full rank such that the torus is equipped with a polarization $Q$. I want to see an explicit degeneration which gives the tropical abelian variety.

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  • $\begingroup$ For $g=1$, the idea is to express your elliptic curve as $\mathbb{C}^*/q^{\mathbb{Z}}$. $\endgroup$
    – Nulhomologous
    Commented Nov 23 at 10:48
  • $\begingroup$ @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., $g=1$? $\endgroup$
    – divergent
    Commented Nov 23 at 13:00
  • $\begingroup$ The problem here is that it is not true that a tropical abelian variety is a degeneration of a complex family: it is true in some sense over fields complete with respect to a non-archimedean valuation. Or if you change what it means "to degenerate". $\endgroup$
    – Nulhomologous
    Commented Nov 26 at 7:57
  • $\begingroup$ About the case $g=1$, it is just to simplify. Any complex abelian variety can be express as $(\mathbb{C}^*)^g/\Lambda$ for some "lattice" $\Lambda$ (plus a polarization), by esentially taking $\exp(2\pi z)$ to the usual expression. $\endgroup$
    – Nulhomologous
    Commented Nov 26 at 13:10
  • $\begingroup$ @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? $\endgroup$
    – divergent
    Commented Nov 27 at 13:09

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