Timeline for Moduli space of complex and anti-complex tori?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 18 at 13:48 | comment | added | psl2Z | Since for $d=1$ it is $\mathbb{R}^*SO(2)\backslash GL_2(\mathbb{R})/GL_2(\mathbb{Z}) \cong \mathbb{C}$ and $\mathbb{R}^*O(2)\backslash GL_2(\mathbb{R})/GL_2(\mathbb{Z}) \cong \{\text{Im}(z) \geq 0\}$. | |
| Oct 18 at 13:44 | comment | added | psl2Z | Ok. I was first confused and thought that $I$ was the indentity. But $I$ is the complex structure. In the case that $d$ is even, doesn't $I \mapsto -I$ produce the identity on the Teichmüller space and the moduli spaces of complex and of complex/anticomplex structures are thus equal in the case $d$ even? In the case $d$ odd $I \mapsto -I$ produces an involution on the Teichmüller space? For example in case $d = 1$ it gives $\cong \mathbb{C}$ for the moduli space complex structures and $\cong \{\text{Im}(z) \geq 0\}$ for the moduli space of complex/anticomplex structures? | |
| Oct 11 at 21:14 | comment | added | Misha Verbitsky | Complex structure on a vector space defines the orientation. Two components are complex structures with the opposite orientation. The correspondence $I \mapsto -I$ identifies these two components when dimension is odd, and produces an involution on the Teichmuller space of complex structures when it is even. | |
| Oct 5 at 1:03 | history | edited | LSpice | CC BY-SA 4.0 |
Fixed typo (sorry, didn't get it in the last edit)
|
| Oct 4 at 23:12 | comment | added | psl2Z | In particular all moduli spaces for $d \geq 2$ would simply be homeomorphic to $(\mathbb{R},\tau)$ with the trivial topology $\tau$? | |
| Oct 4 at 23:05 | comment | added | psl2Z | Thank you for the reference. It looks interesting. What do you mean by "these two components"? The two sheets of the "covering"? And what do you mean by "orientation"? I think if we had real dimensions then $-I$ would induce the opposite orientation if and only if $d$ was odd? Further, then it is not interesting/ useless to study the moduli space for $d \geq 2$ topologically, i.e. only the $1$-dimensional case gives an interesting moduli space? | |
| Oct 4 at 22:23 | history | edited | LSpice | CC BY-SA 4.0 |
`\dotsc` and `\DeclareMathOperator`
|
| Oct 4 at 22:10 | history | answered | Misha Verbitsky | CC BY-SA 4.0 |