Moser's trick

Let $M$ be compact and let ${\omega }_t$ be a smooth family of symplectic forms with constant de Rham cohomology class. Choose ${\sigma }_t$ smoothly so that ${\dot{\omega }}_t=d{\sigma }_t$, solveand let ${\varphi }_t$ be the flow of $X_t$. Thenso ${\varphi }_t^{\ast }{\omega }_t={\omega }_0$.

Any two smooth degree-$d$ hypersurfaces in ${\mathbb{CP}}^n$, equipped with the restricted Fubini-Study form, are symplectomorphic. Join them through the connected complement of the discriminant in the parameter space, use the resulting smooth family to identify the fibers, and apply Moser's trick to the cohomologous restricted forms.

The degree-$d$ Fermat hypersurface in ${\mathbb{CP}}^n$ is

The Fermat hypersurface $z_0^d+\cdots +z_n^d=0$ is preserved by multiplying each coordinate by a $d$th root of unity. Projective scalar multiplication is trivial, so the effective diagonal group isThese transformations preserve the Fubini-Study form.