If there are 2-forms $\omega_1$ and $\omega_2$ in the manifold $M_1$ and $M_2$ respectively, does saying something like $[\omega_1]=[\omega_2]$ make sense if $M_1$ and $M_2$ don't have the same almost-complex structure, where $[\omega_1]$ denotes the cohomology class of $\omega_1$?
Additional info: We have $(M,\omega,J_1)$ and $(M,\omega,J_2)$, where $J_i$ is the almost-complex structure and $J_1$ and $J_2$ are homotopic by a 1-parameter family of diffeomorphisms $\varphi_t:M\rightarrow M$. The homotopy does not necessarily preserve $J_1$. $\omega$ is symplectic and $J_i$ is $\omega$- tame. Then we blow-up symplectically, and obtain respectively $\tilde{\omega}_1,\tilde{J}_1$ and $\tilde{\omega_2},\tilde{J}_2$, where $\tilde{J}_i$ is $\tilde{\omega}_i$-tame. Why are these two 2-forms not necessarily cohomologous?
