It is well-known that modular curves parametrize elliptic curves with level structures. For the purpose of this question, I will work complex-analytically and describe analytically the moduli space $$\operatorname{SL}(2,\mathbb{Z})\backslash \mathcal{H}$$ which classify complex elliptic curves in somewhat naive sense:
Think of complex elliptic curves as complex tori $\mathbb{C} / \Lambda$. One may write every lattice $\Lambda$ in the form $\Lambda_\tau:=\mathbb{Z} \tau \oplus \mathbb{Z}$ for some $\tau \in \mathcal{H}$; and two complex tori are isomorphic iff the underlying lattices are homothetic, that is, differ by a scaling. In terms of $\Lambda_\tau$, we have $\Lambda_\tau \simeq \Lambda_{\tau^\prime}$ iff $\tau^\prime = \gamma \tau$ for some $\gamma \in \operatorname{SL}_2(\mathbb{Z})$. This is how I first encountered the moduli nature modular curves (of course algebraic geometry provides a much more beautiful and powerful way to understand this phenomenon).
Now, I would like to regard modular curves as moduli spaces of “Hodge structures on the (real) torus”. (let me postpone what I mean by Hodge structures below). The previous process is like varying the complex structures on the torus by varying the underlying lattice; and I want to do the opposite way, namely, fixing $\Lambda = \mathbb{Z}^2$ inside $\mathbb{R}^2$ and try to put different complex structures on $\mathbb{R}^2 / \mathbb{Z}^2$. Obviously, we should end up with the same object $\operatorname{SL}(2,\mathbb{Z})\backslash \mathcal{H}$ as before. To me, however, this seems a lot less direct and has been bothering me for a while. My attempt:
A. Complex structures
- A complex structure on a real vector space $V$ is a real-linear endomorphism $J: V \to V$ satisfying $J^2 = - \mathrm{Id}$.
- With such a $J$ on $V$, $V$ can be endowed with a complex vector space structure: $(x+iy) v := xv + yJv$ for all $z = x+iy \in \mathbb{C}$ and $v \in V$.
- Two complex structures $J_1$ and $J_2$ on $V$ are isomorphic if there exists an endomorphism $\Phi$ of $V$ such that $\Phi \circ J_1 = J_2 \circ \Phi$.
- Same construction works for vector bundles on a real smooth manifold.
Consider first the case where $V = \mathbb{R}^2$ is a real vector space. I claim that the set of all complex structures on $\mathbb{R}^2$ can be written as $$\operatorname{GL}_2(\mathbb{R})/\mathbb{C}^\times.$$
To see this, note that every such a matrix $J$ can be written as $A \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} A^{-1}$ for some $\operatorname{GL}_2(\mathbb{R})$, and so the set of complex structures on $V$ is in 1-1 correspondence with $\operatorname{GL}_2(\mathbb{R}) \cdot \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} / \text{Stabilizer}$, and then we check that actually $\text{Stab} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ are matrices of the form $\begin{pmatrix} a & b \\ -b & a \end{pmatrix}$ which as a set $\simeq \mathbb{C}^\times$.
These complex structures on $V = \mathbb{R}^2$ are all isomorphic to each other. Indeed, let $J_1$ and $J_2$ be two complex structures on $V$, then $J_2$ can be written as $A J_1 A^{-1}$ for some $A$ for some $A \in \operatorname{GL}_2(\mathbb{R})$. Viewing $A$ as an endomorphism $\Phi$ on $V$ shows that $J_1$ and $J_2$ are isomorphic. Thus, up to isomorphism, there is only one complex structure on $\mathbb{R}^2$ as a real vector space; and pedantically $$ \operatorname{GL}_2(\mathbb{R}) \backslash \operatorname{GL}_2(\mathbb{R})/\mathbb{C}^\times = \{1\}. $$
However, I think there are two complex structures on the real manifold $\mathbb{R}^2$: In addition to the usual complex structure such that $\mathbb{R}^2 \simeq \mathbb{C}$ (as complex manifold), the open unit disk is homeomorphic to $\mathbb{R}^2$, and hence we may equip $\mathbb{R}^2$ with a complex structure such that $\mathbb{R}^2 \simeq \mathbb{D}$ (the Poincare upper half-plane). Conversely, Riemann’s uniformization theorem tells us that every simply connected complex surface is biholomorphic to either $\mathbb{C}, \mathbb{D}$, or the Riemann sphere $\mathbb{S}^2$.
I guess the point is, when regarding $\mathbb{R}^2$ as a manifold, the $J$ matrix at different points may vary in a smooth way. What we really do is to first give a complex structure on the tangle bundle (the so-called “almost complex structures”), and then “descend” it to a complex structure on the manifold. This is doable if the almost complex structure on the tangent bundle is “integrable”, which is always the case for surfaces by a great theorem proved in 1957 due to August Newlander and Louis Nirenberg.
Finally, we get complex structures on the torus $\mathbb{R}^2/\mathbb{Z}^2$ by first varying complex structures on the manifold $\mathbb{R}^2$ and then passing to on the quotient manifold. The matrices in $\operatorname{GL}_2(\mathbb{R})$ that preserve the lattices $\mathbb{Z}^2$ are precisely $\operatorname{GL}_2(\mathbb{Z})$. Hence, we quotient by $\operatorname{GL}_2(\mathbb{Z})$ and classify complex structures on the torus by $$ \operatorname{GL}_2(\mathbb{Z})\backslash \operatorname{GL}_2(\mathbb{R})/ \mathbb{C}^\times \simeq \operatorname{SL}(2,\mathbb{Z})\backslash \mathcal{H} $$
However, implicitly we are assuming that the $J$ matrices at different points are the same, i.e., ignoring the case where $\mathbb{R}^2$ is equipped with the hyperbolic complex structure. This is one issue I have yet to understand.
B. Hodge structures
- An integral Hodge structure on a free abelian group $V_\mathbb{Z}$ is a decomposition $V_\mathbb{C}$ (complxification of $V$) into $\oplus_{p,q} V^{p,q}$ where $V^{p,q}$ are $\mathbb{C}$-subspaces of $V_\mathbb{C}$ and $V^{q,p} = \overline{V^{p,q}}$.
- We define the type of $V$ to be the set of ${p,q} \in \mathbb{Z}^2$ such that $V^{p,q} \neq 0$.
- When $X$ is a complex Kähler manifold, we take $V_\mathbb{Z} = H^k(X,Z)$ in practice.
Hodge structures of type $\{(1,0), (0,1)\}$ on a real vector space $V$ correspond bijectively to complex structures on $V$: On the one hand, to a complex structure $J$ on $V$ we may associate the decomposition $V_\mathbb{C} = V^{1,0} \oplus V^{0,1}$ where $V^{1,0}$ (resp. $V^{0,1}$) is the $I$- (resp. $-I$-) eigenspace; and on the other, given such a Hodge structure we may consider the element $J_C$ of $\operatorname{GL}(V_\mathbb{C})$ that acts by multiplication by $I$ on $V^{1,0}$ and $-i$ on $V^{0,1}$. Note that $J_\mathbb{C}$ commutes with complex conjugation, it restricts to a map on $V$.
Another missing piece is the correspondence between Hodge structures and complex structures on a complex manifold.
Main questions
- I was told that the torus $\mathbb{R}^2 / \mathbb{Z}^2$ is a parallelizable manifold, namely, all vector fields admit global trivialization. If that’s the case, can I make this argument rigorous? Am I on the right track? If not, is my argument just a “coincidence”?
- It seems that one needs to do a very dangerous thing -- working with the huge diffeomorphism group -- at some point, so my argument might be shaky… Even if this can be made “correct”, I would still complain that it is too special to the modular curve case.
- Would this lead to Deligne’s formalism of Shimura varieties (say, of PEL type)? What am I missing?
- Let me put also what I learned from Professor Martin Orr here in quote as well:
Trying to directly reason about isomorphism classes of complex structures on the torus seems likely to be difficult. So I would want to get around this by proving:
Claim. Every isomorphism class of complex structures on $\mathbb{S}^1 \times \mathbb{S}^1$ contains a complex structure which is compatible with the group structure. Furthermore, if two complex structures $J_1$, $J_2$ on $\mathbb{S}^1 \times \mathbb{S}^1$ which are both compatible with the group structure are isomorphic as complex structures, then a diffeomorphism of $\mathbb{S}^1 \times \mathbb{S}^1$ mapping $J_1$ to $J_2$ is a composition of a Lie group automorphism and a translation.
Then you reduce to classifying complex structures on $\mathbb{S}^1 \times \mathbb{S}^1$ compatible with the group structure, up to isomorphisms of complex structures compatible with the group structure. In my Claim, there are two ways to interpret “compatible with the group structure”, which I think are ultimately equivalent:
(1) The group law is holomorphic with respect to the complex structure. Then you are back to classifying complex elliptic curves.
(2) The complex structure is invariant under translations (a priori weaker than (1)).
To prove the Claim with version (2), you could use an argument relying on Riemann uniformization to show that the pullback of the complex structure to $\mathbb{R}^2$ is isomorphic to the constant one.
