Timeline for Complex torus, C^n/Λ versus (C*)^n
Current License: CC BY-SA 4.0
19 events
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| Aug 21, 2019 at 23:34 | comment | added | reuns | Also I would say a compact complex manifold with some group law given by analytic functions is automatically an abelian group and a complex torus (because the map $Lie(G) \to Lie(G), x \mapsto g^{-1} x g$ makes $g$ an element of $GL(Lie(G))$ and the map $G\to GL(Lie(G))\subset \Bbb{C}^{d^2}$ is analytic and bounded thus constant by the maximum modulus principle) that's why abelian varieties are sometimes defined in some abstract way (complete group scheme...) without mentioning the group law is abelian and the biholomorphism to a complex torus. | |
| Jul 17, 2019 at 11:50 | comment | added | Qfwfq | The funny thing about this terminology is a complex algebraic torus (a.k.a. $(\mathbb{C}^{*})^n$) is not the same thing as an algebraic complex torus (a.k.a. $\mathbb{C}^n/\Lambda\subset \mathbb{CP}^N$) | |
| Jul 17, 2019 at 2:09 | comment | added | Mikhail Borovoi | @JohannesHahn: OP correctly wrote $SU(2,\Bbb C/\Bbb R)$ and $SU(3,\Bbb C/\Bbb R)$. | |
| Jul 17, 2019 at 1:12 | history | edited | Johannes Hahn | CC BY-SA 4.0 |
LaTeXified the answer
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| Jul 16, 2019 at 17:55 | review | Close votes | |||
| Jul 17, 2019 at 0:47 | |||||
| Jul 16, 2019 at 15:53 | answer | added | Bugs Bunny | timeline score: 1 | |
| Jul 22, 2017 at 17:36 | answer | added | user21574 | timeline score: 7 | |
| Aug 13, 2012 at 14:48 | vote | accept | Jack Schmidt | ||
| Apr 6, 2010 at 18:19 | comment | added | Pete L. Clark | As you have seen, the terminology "algebraic torus", though common, can be confusing. It is usually used for an algebraic group which is, over the algebraic closure, isomorphic to the direct sum of n copies of the multiplicative group. "Complex torus" is usually used for a compact, complex Lie group, necessarily then C^n modulo a lattice. But an "algebraic torus over C" is very reasonably called a "complex torus" and many complex tori -- those with Riemann forms, e.g. all elliptic curves -- are algebraic! I prefer the term "linear torus" for the (C^*)^n guy. | |
| Apr 6, 2010 at 16:31 | comment | added | some guy on the street | We're really talking about spaces here, not rings, so if you want to "tensor" something, that's where you'd need to be working. The "$\times R$" you see is akin to the way $\mathbb{C}$ is like $\mathbb{R}\times\mathbb{R}$; and we can talk of analytically continuing functions on the unit circle over the complex torus as well --- this is analogous to analytic continuation of functions on $\mathbb{R}$, but we might think sooner of Laurent series in this new setting, for example | |
| Apr 6, 2010 at 16:30 | answer | added | stankewicz | timeline score: 5 | |
| Apr 6, 2010 at 16:18 | comment | added | Jack Schmidt | Mariano: so projective varieties are compact? C* is like PC^1 minus two points, so very much not compact? SGOTS: thanks. so the jump from flat torus to complex torus is the addition of a "complex structure". How do you think of C* as a complexified torus? To my mind, tensoring R/Z with C gives something quite different. It looks more like R/Z x R, which seems quite strange. | |
| Apr 6, 2010 at 16:10 | comment | added | some guy on the street | Let me try those again... $\mathbb{C}^n/\Lambda$ first; $(\mathbb{C}^*)^n$ second... | |
| Apr 6, 2010 at 16:08 | comment | added | Mariano Suárez-Álvarez | A nice book on all this is Complex tori and Abelian varieties, by Olivier Debarre. There you'll find for example the Riemann conditions which are necessary for a torus to be abelian. | |
| Apr 6, 2010 at 16:08 | comment | added | some guy on the street | $\mathbb{C}^n/Λ$ is an even-dimensional real torus with a specific (flat) complex structure. $\mathbb{C}*^n$ is a complex(ified) torus. | |
| Apr 6, 2010 at 16:07 | answer | added | Simon Rose | timeline score: 13 | |
| Apr 6, 2010 at 16:06 | comment | added | Mariano Suárez-Álvarez | If $k$ is a global field, then the group of $k$-points of an abelian variety is finitely generated. Maybe that's what your memory is recalling. | |
| Apr 6, 2010 at 16:04 | comment | added | Mariano Suárez-Álvarez | An abelian variety is projective, so $\mathbb C^*$, which is not even compact, is not one. | |
| Apr 6, 2010 at 16:01 | history | asked | Jack Schmidt | CC BY-SA 2.5 |