Complex structure of a torus Given the definition of complex structure for a complex manifold:
the real $(1,1)$ type tensor $J_p : T_p M \rightarrow T_p M $ defined by 
$$ J_p \left(\frac{\partial}{\partial x^\mu} \right) = \frac{\partial}{\partial y^\mu} \qquad J_p \left(\frac{\partial}{\partial y^\mu} \right) = - \frac{\partial}{\partial x^\mu} $$ 
how can I "see" how it is related to the modular parameter $\tau$ of a $T^2$ for example? I mean, why is $\tau$ said to correspond to the complex structure of $T^2$?
 A: Ok, after a more careful re-reading of Nakahara, I think I got it:
the modular parameter defines the complex structure of $T^2$, since $T^2$ may carry a number of (different) complex structure.
$T^2$ is (homeomorphic to) the manifold $\mathbb{C}/L(\omega_1, \omega_2)$, where $L(\omega_1, \omega_2) \equiv \{ \omega_1 m + \omega_2 n \ | \ m,m \in \mathbb{Z} \}$ is the lattice defined by two non-vanishing complex numbers $\omega_1, \omega_2$. As a consequence, the complex structure of the covering $\mathbb{C}$ induces that of $\mathbb{C}/L(\omega_1, \omega_2)$ (i.e. the torus), which is then a complex manifold (this can be proven more in general). In particular, the complex structure of this manifold is defined by the choice of $\omega_1, \omega_2$. Different $\omega_1, \omega_2$ correspond (in principle) to different lattices, i.e. to different ways of identifying points in $\mathbb{C}$, i.e. to different ("structures" for) $T^2$. It turns out that some of these (a priori) different lattice (and then $T^2$) are equivalent. This happens when $(\omega_1, \omega_2)$ are multiplied by a constant complex number or transformed under $PSL(2,\mathbb{Z})$. Therefore, a $T^2$ defined by $(\omega_1, \omega_2)$ is equivalently described by $(1, \tau)$, with $\tau \equiv \omega_2/\omega_1 \in H/PSL(2,\mathbb{Z})$ (see Nakahara for details). 
This is why $\tau$ defines the complex structure of $T^2$.
