Suppose $\omega$ is a real closed $(1,1)$ form on a compact Kähler manifold. If we have a real $d$closed two form $\sigma$ such that $[\sigma]=[\omega] \in H^2(M)$, can we claim that this two form $\sigma$ is also a $(1,1)$ form?
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I think not. Note that replacing $\sigma$ by $\sigma  \omega$ reduces us to the case $\omega=0$. Your question in that case is whether every real $d$exact 2form is a $(1,1)$form. Now unless I'm mistaken, on projective space $\mathbf{CP}^2$ with homogeneous coordinates $(a:b:c)$, we get a real $d$exact 2form with components of all types ($(2,0)$, $(1,1)$, $(0,2)$) by putting $$ \sigma =d\left(\frac{A}{A+B+C} d\left(\frac{B}{A+B+C}\right)\right) =\frac{AdB\wedge dC+BdC\wedge dA+CdA\wedge dB}{(A+B+C)^3} $$ where $(A,B,C)=(a^2, b^2, c^2)$. 

