Orthogonal symplectic classes with respect to intersection product

Question

Let $M$ be a simply connected smooth compact four manifold. Now I am trying to find an example such that

(1) there is two symplectic forms $\omega$ and $\sigma$ on $M$, and

(2) the intersection product $[\omega] \cdot [\sigma]$ is zero.

Obviously, if we drop the condition "simply connectedness", then four-dimensional compact torus $T^4$ satisfies the above condition. (Set $\omega = dx_1 \wedge dx_2 + dx_3 \wedge dx_4$ and $\sigma = dx_1 \wedge dx_3 + dx_4 \wedge dx_2.$) Also, it is not hard to show that there is no such example if $M$ is simply connected and $b_2^+(M) = 1$.

How about the case when $b_2^+(M) > 1$?

Thank you in advance.

Answer 1

Take for $M$ a K3 surface; there is a holomorphic 2-form $\varphi $ on $M$ which is everywhere nonzero. Put $\omega =\varphi +\bar{\varphi }$, $\sigma =i(\varphi -\bar{\varphi })$. These 2-forms are real and symplectic since $\omega \wedge \omega =\sigma \wedge \sigma = 2 \varphi \wedge \bar{\varphi }$ is everywhere nonzero, and $[\omega ]\cdot [\sigma ]=0$.