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 ${\varphi }_t^{\ast }{\omega }_t={\omega }_0$.

Two closed symplectic forms agreeing as ambient bilinear forms along a compact embedded submanifold are equivalent near it by a diffeomorphism fixing it pointwise. Shrink until their affine interpolation is nondegenerate. A relative homotopy primitive $\eta$ for their difference vanishes as an ambient covector on the submanifold. Solving ${\iota }_{Z_t}{\omega }_t=-\eta$ gives a vector field vanishing there; its local flow fixes the submanifold and makes ${\omega }_t$ constant under pullback.

On a compact connected smooth manifold without boundary, two positive volume forms for the same orientation are related by an orientation-preserving diffeomorphism exactly when their total integrals agree. A positive interpolation and Cartan's magic formula give the smooth isotopy realizing the equality.

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.