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I am beginning to read Birkenhake-Lange, Complex Abelian Varieties, where they define a complex torus as being

a quotient $X=\mathbb{C}^g/\Lambda$, where $\Lambda$ is a lattice in $\mathbb{C}^g$. This is a complex manifold of dimension $g$. It inherits the structure of a complex Lie group from the vector space $\mathbb{C}^g$.

Then, they use freely the notion of "subtorus" without defining it explicitly.

What is meant by subtorus ? A subgroup of $X$ isomorphic (as a group) to a complex torus ? A Lie subgroup of $X$ ?

If $\Lambda'$ is a subgroup of $\Lambda$ of rank $2g'$ such that $V'=\Lambda\otimes\mathbb{R}$ is a complex vector subspace of $V$, then $V'/\Lambda'$ is a complex torus. In exercise 1, they seem to say that this can be realized as a subtorus of $X$. How ?

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  • $\begingroup$ In some other notes, they seem to claim that any subtorus is of the form $(V+\Lambda)/\Lambda$, where $V$ is a subspace of $\mathbb{C}^g$. $\endgroup$
    – Klaus
    Mar 28, 2014 at 17:27

1 Answer 1

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A subtorus of $X=\mathbb{C}^g/\Lambda$ is a quotient $W/(\Lambda\cap W)$, where $W$ is a complex vector space of $\mathbb{C}^g$ and $\Lambda\cap W$ is a lattice in $W$ (careful, $\Lambda\cap V$ is rarely a lattice in $V$ for an arbitrary complex vector subspace $V$). You could also think of a subtorus as a closed immersion of a torus $Y$ into $X$ that is also a group homomorphism. These are equivalent definitions.

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  • $\begingroup$ So subtori are Lie subgroups of the Lie group $X$, right ? $\endgroup$
    – Klaus
    Mar 29, 2014 at 16:12
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    $\begingroup$ Yes, in particular they are Lie subgroups. Specifically, complex subtori correspond to connected compact complex Lie subgroups of your original torus. $\endgroup$
    – rfauffar
    Mar 29, 2014 at 23:13
  • $\begingroup$ Thank you, that's much clearer now! $\endgroup$
    – Klaus
    Mar 29, 2014 at 23:23

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