Suppose $T= \mathbb{C}^n/\Gamma$ is a $n$-dim complex torus.
How to prove that every exact $2$-form which has no $(0,2)$ component must be the image of a $(1,0)$ form?
Is every torus Kahler?
If the torus is Kahler, can we use hodge decomposition to prove it?
Every torus are Kahler: the standard form $\omega = \frac{\sqrt{-1}}{2}\sum_{i=1}^n dz^i \wedge d\bar z ^i$ descends to a Kahler form on the torus. Let $\alpha$ be a 1 form. Write
$$\alpha = \alpha^{1,0} + \alpha^{0,1}$$
so $$d \alpha = d\alpha^{1,0} + d\alpha^{0,1}$$
If $d\alpha$ has no $(0,2)$ component, then $d\alpha^{0,1}$ is a $(1,1)$ form and the $\partial \bar\partial$-lemma tells us that there is a function $\phi$ such that
$$d\alpha = \partial \bar\partial \phi = - \bar \partial \partial \phi = -d \partial \phi$$
Thus
$$d\alpha = d\big(\alpha^{1,0} - \partial \phi\big)$$
and $\alpha^{1,0} - \partial \phi$ is a $(1,0)$ form.