Let $a \in H^2(M, \mathbb{R})$ be a cohomology class of a closed manifold $M$ of dimension $2n$. For the cohomology class $a$ to represent a symplectic form on $M$, we must have $a^n \neq 0$. This is not sufficient in general as we see in the case $M = C\mathbb{P}^2 \# C\mathbb{P}^2$.
Let's consider the case $M = \Sigma_{g_1} \times \Sigma_{g_2}$ where $\Sigma_{g_i} $ is a compact Riemann surface of genus $g_i \geq 2$. This manifold $M$ admits a product symplectic form. I wonder whether there are "exotic" symplectic forms on $M$.
We can find a pair of cohomology classes $x_i, y_i \in H^1(\Sigma_{g_i}, \mathbb{R})$ with $x_i y_i \neq 0$. If we take
$a = \pi_1^*x_1 \cup \pi_2^*x_2 + \pi_1^*y_1 \cup \pi_2^*y_2 \in H^2(M, \mathbb{R})$,
where $\pi_i$ is the projection to the $i$th factor, does $a$ represents a symplectic form on $M$?
