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I want to show that the complex torus $\mathbb{T}= \mathbb{C}/\Lambda$ is a compact, connected complex manifold of complex dimension $1$. $\Lambda$ is a discrete subgroup of $\mathbb{C}$ generated by two complex numbers $w_1$ and $w_2$ that are linearly independent over $\mathbb{R}$. The equivalence relation on $\mathbb{C}$ is defined as $z\sim z' \iff z-z' \in \Lambda.$

  1. If I can explicitly construct a homeomorphism from $\psi: \mathbb{T} \rightarrow S^1\times S^1 \subset \mathbb{R}^4$, I can say that:

    • $\mathbb{T}$ is compact as $S^1\times S^1$ is cartesian product of compact sets.

    • $\mathbb{T}$ is connected as it is relatively easy to to show that $S^1\times S^1$ is path-connected (by traversing arcs) hence connected.

    • $\mathbb{T}$ is Hausdorff and second countable as $S^1\times S^1 \subset \mathbb{R}^4$ is so as a subspace.

      I need help with explicitly finding $\psi$.

  2. How can I construct the holomorphic coordinate covering of $\mathbb{T}$? I can take open disks in $\mathbb{C}$ with sufficiently small radius $\epsilon$ such that there is at most one point of intersection between the disk and any equivalence class. But, what are the local holomorphic coordinates?

テレビ スクリーン
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    That torus has the quotient topology and $\mathbf C \to \mathbf C/\Lambda$ is continuous and surjective, so $\mathbf C/\Lambda$ is connected since $\mathbf C$ is connected. For compactness, find a compact subset of $\mathbf C$ whose image is all of $\mathbf C/\Lambda$. – KCd Jan 27 '21 at 19:13
  • So the connectedness of the torus can follow from the connectedness of $\mathbb{C}$. A compact subset can be taken as one segment of the lattice. Hausdorff and second countable properties follow similarly. And I wouldn't need such homeomorphism $\psi.$ Do you have anything about 2.? – テレビ スクリーン Jan 27 '21 at 19:17
  • See https://math.stackexchange.com/questions/496571/under-what-conditions-the-quotient-space-of-a-manifold-is-a-manifold. – KCd Jan 27 '21 at 19:20
  • Are you referring to the accepted answer and the quoted theorem "Lee's Introduction to Smooth Manifolds, Theorem 9.19" stated in terms of smooth manifolds? How would I translate that to complex manifolds? How can I go about $\Lambda$ acting smoothly, freely, properly on $\mathbb{C}$? – テレビ スクリーン Jan 27 '21 at 19:31
  • I don't want to solve every part for you. Use that stackexchange link as motivation. If you genuinely get stuck, you can find a discussion of how to make a torus $\mathbf C/\Lambda$ into a complex manifold in books on Riemann surfaces. – KCd Jan 27 '21 at 20:16
  • The following file was really helpful. Its using the fact $\Lambda$ acting "evenly" and the tool covering spaces. I think it is very similar to that answer you link to @KCd, thanks a lot for your help. – テレビ スクリーン Jan 27 '21 at 20:20

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