Taubes–Seiberg–Witten theory [@TAUBES5].

[**3.**]{} $M$ admits an orientation preserving diffeomorphism that preserves the fibration and reverses the orientation of the fiber, and an orientation reversing diffeomorphisms that preserves the fibration and the orientation of the fiber. Thus every symplectic form on $M$ is diffeomorphic to one that is compatible with the fibration and orientations.

[**4.**]{} A theorem of Lalonde–McDuff [@LM1; @LM2] asserts that any two symplectic forms ${{\omega}}_0,{{\omega}}_1$ on $M$ that are compatible with the fibration and orientations can be joined by a path of symplectic forms. They also proved that the path can be chosen in the same cohomology class when $[{{\omega}}_0]=[{{\omega}}_1]$. Thus $$\label{eq:LM}
{{\omega}}_0\stackrel{(\mbox{\tiny{a}})}{\sim}{{\omega}}_1\qquad\iff\qquad
{{\omega}}_0\stackrel{(\mbox{\tiny{b}})}{\sim}{{\omega}}_1\;\;\mbox{ and }\;\;[{{\omega}}_0]=[{{\omega}}_1]$$ Here (a) and (b) denote the equivalence relations in \[para:equiv\].

[**Conclusion.**]{} [*On a ruled surface any two symplectic forms are deformation equivalent, and diffeomorphic if they represent the same cohomology class. In the latter case they are homotopic if and only if they are isotopic.*]{}

\[ex:CP21\]Let $$M := {{{{\mathbb C}}\mathrm{P}}}^2\#\overline{{{{{\mathbb C}}\mathrm{P}}}}^2.$$ Let $L\in H_2(M;{{\mathbb Z}})$ be the homology class of the line in ${{{{\mathbb C}}\mathrm{P}}}^2$ and $E\in H_2(M;{{\mathbb Z}})$ be the class of the exceptional divisor, both with their complex orientations. They have self-intersection numbers $L\cdot L=1$ and $E\cdot E=-1$. Then 