Moduli Spaces of Higher Dimensional Complex Tori

Question

I know that the space of all complex 1-tori (elliptic curves) is modeled by $SL(2, \mathbb{R})$ acting on the upper half plane. There are many explicit formulas for this action.

Similarly, I have been told that in the higher dimensional cases, the symplectic group $Sp(2n, \mathbb{R})$ acts on some such space to give the moduli space of complex structures on higher dimensional complex tori. Is there a reference that covers this case in detail and gives explicit formulas for the action?

In the 1-dimensional case, all complex tori can be realized as algebraic varieties, but this is not the case for higher dimensional complex tori. Does the action preserve complex structures that come from abelian varieties?

Crossposted at https://math.stackexchange.com/questions/345713/moduli-spaces-of-higher-dimensional-complex-tori where it has been unanswered for awhile.

Answer 1

The fact that all $1$-dimensional tori are projective means care is sometimes needed in making analogies with higher dimensional tori. This is one of those times. The natural 'moduli space' of all $d$-dimensional complex tori constructed by Will is not very nice when $d>1$. For example, the action of $GL_{2d}(\mathbb Z)$ on $GL_d(\mathbb C)\backslash GL_{2d}(\mathbb R)$ isn't properly discontinuous, so the resulting quotient space isn't Hausdorff (an observation due to Siegel).

A slightly better state of affairs is available if we begin by looking at the Siegel upper half-plane $$ \mathcal H_d = \{ \tau \in M_d(\mathbb C) \colon \tau^t = \tau, \mathrm{Im}(\tau) > 0 \}. $$ The group $Sp(2d,\mathbb R)$ acts transitively on $\mathcal H_d$ via $$ \begin{pmatrix}A & B\\C & D\end{pmatrix} \tau = (A\tau+B)(C\tau+D)^{-1} $$ (this is well-defined) and the isotropy subgroup of $iI_d$ is (essentially) $U(g)$. Thus we can view $\mathcal H_d$ as $Sp(2d,\mathbb R)/U(d)$. Now a point $\tau \in \mathcal H_d$ gives us a complex torus $\mathbb C^d/(\mathbb Z^d + \mathbb Z^d \tau)$ -- but this torus is not an arbitrary one: it comes with a "principal polarization". Moreover the natural action of $Sp(2d,\mathbb Z)$ on $\mathcal H_d$ (which, by the way, is properly discontinuous) preserves the isomorphism class of the corresponding torus as well as its principal polarization. Thus the points of the space $\mathcal H_d/Sp(2d,\mathbb Z)$ are in bijection with isomorphism classes of principally polarized $d$-dimensional complex tori (these are abelian varieties). Note that if $d=1$ we're reduced to the familiar modular curve $\mathcal H/SL(2,\mathbb Z)$ which parametrizes all $1$-dimensional tori.

For (much) more on this, see II.4 of Mumford's Tata Lectures on Theta I or van der Geer's chapter in The 1-2-3 of Modular Forms.

Answer 2

A complex torus of dimension $d$ can be written as a quotient $\mathbb C^d/\mathbb Z^{2d}$. Thus it is determined by a map $\mathbb Z^{2d} \to \mathbb C^{d} \cong \mathbb R^{2d}$. We can specify this with an element of $GL_{2d} (\mathbb R)$.

However, sometimes two elements of $GL_{2d}(\mathbb R)$ give the same complex torus. This can happen in two ways. Either we can reparametrize $\mathbb C^d$, via an element of $GL_d(\mathbb C)$, or we can reparametrize $\mathbb Z^{2d}$, via an element of $GL_{2d} (\mathbb Z)$. Thus, the moduli space of complex tori is $GL_d(\mathbb C) \backslash GL_{2d}(\mathbb R) / GL_{2d}(\mathbb Z)$.

Riemann proved that a torus is an abelian variety if and only if it has a Riemann form. One can construct complex tori that do not have Riemann forms. If one is interested in abelian varieties, the moduli spaces that are best-behaved are usually the moduli space of abelian varieties with a fixed Riemann form. One can classify these by fixing a Hermitian form on $\mathbb C^d$ and associated symplectic form. Similarly, fix a symplextic form on $\mathbb Z^{2d}$. Then the embedding $\mathbb Z^{2d} \to \mathbb C^d$ must be symplectic, corresponding to an element of $SP_{2d}(\mathbb R)$. The reparamaterization on the complex side now comes from an element preserving the Hermitian form, meaning an element of $U_d(\mathbb C)$. On the integer side, it is an element of $SP_{2d}(\mathbb Z)$. This gives the double coset description $U_d(\mathbb C)\backslash SP_{2d}(\mathbb R)/SP_{2d}(\mathbb Z)$.

I think that is the source of the "action of $SP_{2d}(\mathbb R)$" you are looking for.