forms (after rescaling) are in rational cohomology classes. His techniques allowed him to obtain some lower and upper bounds for $N_{{\rm stab}}(M^{4},\omega)$ which can be explicitly computed in some cases. In particular, he showed that $N_{{\rm stab}}({\mathbb{CP}}^{2})\leq 9$ which, in view of McDuff and Polterovich results, is sharp.

The same techniques apply to rational ruled symplectic $4$-manifolds. Recall that, after rescaling, any such manifold is symplectomorphic to either

-   the trivial bundle $M_{\mu}^{0} := (S^2\times S^2, {\omega}^0_{\mu})$, where the symplectic area of the a section $S^2\times\{*\}$ is $\mu\geq 1$ and the area of a fiber $\{*\}\times S^{2}$ is 1; or

-   the non trivial bundle $M_{\mu}^{1} := (S^{2}\ltimes S^{2},{\omega}^{1}_{\mu})$, where the symplectic area of a section of self-intersection $-1$ is $\mu>0$ and the area of a fiber is $1$.

In [@Bi] Biran showed that $$\label{birdiotriv}
p_k(M_{\mu}^0) =\min\left\{ 1,\frac{k}{2 \mu}\inf\left(\frac{\mu n_1 + n_2}{2 n_1 +2 n_2 -1}\right)^{2}\right\}$$ where the infimum is taken over all naturals $n_1, n_2$ for which the Diophantine equations $$\label{birdioeq}
2 n_1 n_2 =\left(\sum_{i=1}^{i=k}m_i^2 \right)-1~, \qquad 2 (n_1 + n_2) =\left(\sum_{i=1}^{i=k}m_i\right) +1$$ admit a vector solution $(m_1,\ldots, m_{k})\in {\mathbb{N}}^{k}$, while $$p_k(M_{\mu}^1) =
\min\left\{ 1,\frac{k}{2\mu+1}\inf\left(\frac{\mu n_1 + n_2}{n_1 +2 n_2 -1}\right)^{2}\right\}$$ where the infimum is taken over all naturals 