What exactly is the Kahler class of a torus? Is there an intuitional way to understand what the Kahler class of $T^2$ actually is? It would be extremely useful to me if you could provide me some intuition behind it!
 A: Just so this has an answer: Fixing a Kähler class on a compact Riemann surface amounts to fixing the overall area, with some fine print. (Similarly, fixing a Kähler class on a manifold of higher dimension fixes the areas of $2$-dimensional homology classes.)

In more detail, a torus $T$[1] admits a unique flat Kähler metric $g$ of unit area.[2] Let $\omega$ denote the associated Kähler form. 
If $g'$ is an arbitrary Kähler metric of area $\alpha > 0$ on $T$, then the Kähler form $\omega'$ is cohomologous to $\alpha\omega$; that is, $[\omega'] = [\alpha\omega]$ in the de Rham space $H^{2}(T, \mathbf{R}) \simeq \mathbf{R}$.
Thus, a choice of Kähler class on $T$ amounts to fixing the area of $T$.[3] 
[1] Namely, a compact Riemann surface of genus one, a.k.a a holomorphic quotient $\mathbf{C}/\Lambda$ for some rank-$2$ integer lattice $\Lambda$.
[2] This metric is induced (up to overall scale) by the Euclidean metric on $\mathbf{C}$, and is a Riemannian product of circles if and only if $\Lambda$ is rectangular. ("Usually not.")
[3] With the understanding that speaking of $T$ itself entails fixing a holomorphic structure. Two flat tori of equal area are not isometric unless their underlying holomorphic structures are the same. ("Usually not.")
A: See my answer In MO
https://mathoverflow.net/questions/178494/all-kähler-metrics-on-a-complex-manifold/273696#273696
I give a general view of the space of Kahler metrics In the Kahler class $[\omega]$ on a compact Kahler manifold by using Semme's construction which is less known
The Kahler metrics In the Kahler class $[\omega]$  on a compact Kahler manifold $M$ is one-to-one correspondance with exact Lagrangian symplectic submanifolds In the space $\mathcal W_{[\omega]}$.
Now I define the space $\mathcal W_{[\omega]}$.
Let $\{ U_i , i\in I\}$ be a covering of $M$ such that $ \omega|_{U_i}=\sqrt{-1}\partial\bar\partial \rho_i $ . For any $x=y\in U_i \cap U_j$ we identify $(x,v_i)\in T^*U_i $ with $(y,v_j)\in T^*U_j$ if $v_i=v_j+\partial(\rho_i-\rho_j)$ .   Then  $\mathcal W_{[\omega]}$ consists of all these équivalence classes of $[x, v_i]$
This is knows as Semme's construction , see p.12 of the Well written  paper of Tian with Chen
Chen, X. X.; Tian, G.
Geometry of Kähler metrics and foliations by holomorphic discs
Publications Mathématiques de l'IHÉS, Tome 107 (2008) , p. 1-107
http://www.numdam.org/item/PMIHES_2008__107__1_0
The space of Kahler forms in the class of $[\omega]$, can be written as $$\mathcal K=\{\omega_\varphi=\omega+dd^c\varphi, \;\omega_\varphi>0 \}$$
By using Moser lemma, for every $\omega_\varphi\in \mathcal K$, we have a diffeomorphism $F_\varphi^*\omega_\varphi=\omega$, as an exercise we can show that maximal leaf of the involutive distribution of $F$  passing through the complex structure $J$ is the image of the map $$\mathcal K\times G\to F, \; \; \; (\omega_\varphi,\sigma)\to \sigma^* F_{\varphi}^* J$$ where here $G$ is the group of symplectomorphisms.
Note that in Fano Kahler-Einstein manifolds the space of Kahler metrics $\mathcal K$, corresponds to symmetric space $G^\mathbb C/G$.
