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Moser's trick turns variation of symplectic forms into an equation for a time-dependent vector field. Let ${\omega }_t$, $0\le t\le 1$, be a smooth path of symplectic forms on a compact manifold without boundary, with a constant de Rham cohomology class. Choose a smooth family of one-forms ${\alpha }_t$ such that ${\dot{\omega }}_t=d{\alpha }_t$. Such a smooth choice can be made using a fixed auxiliary metric; the essential requirement is this exactness throughout the path.

Nondegeneracy uniquely determines $X_t$ byLet $f_t$ be its flow, with $f_0=\mathrm{id}$. Compactness ensures existence over the whole parameter interval. By Cartan's magic formula,ThusThe path is made constant by a diffeomorphism moving with $X_t$. Every interpolating form must be nondegenerate; equal endpoint cohomology alone does not ensure that every linearly interpolated form is a symplectic form. On a noncompact manifold one instead needs completeness of this flow, or restricts to a sufficiently small neighborhood, as in the local argument below.

The Symplectic Darboux theorem says that every point of a $2n$-dimensional symplectic manifold has local coordinates $(q^1,\dots ,q^n,p_1,\dots ,p_n)$ in whichFirst choose linear coordinates at the point so the form there is standard. The required symplectic basis can be constructed inductively: choose $e,f$ with $\omega (e,f)=1$, split off their span, and repeat on its nondegenerate symplectic orthogonal complement. Extend these coordinates to a chart centered at zero.

Let ${\omega }_0$ be the constant standard form in this chart and $\eta =\omega -{\omega }_0$. The closed form $\eta$ vanishes at zero. On a small star-shaped ball the radial Poincaré lemma supplies a primitiveSince ${\eta }_0=0$, this primitive is $O(|x{|}^2)$. The interpolating forms ${\omega }_t={\omega }_0+t\eta$ are nondegenerate on a common smaller ball, because they all agree with ${\omega }_0$ at zero and $t$ ranges over a compact interval.

Apply the local version of Moser's trick: solve ${\iota }_{X_t}{\omega }_t=-\alpha$. The vector fields are $O(|x{|}^2)$ and fix zero. On a sufficiently small ball their flows exist for $0\le t\le 1$ and remain inside the coordinate chart; the quadratic bound makes their displacement smaller than the available margin. The same pullback calculation gives $f_1^{\ast }\omega ={\omega }_0$. Thus $f_1$ is a local symplectomorphism from the standard ball into $M$, and its inverse supplies the desired Darboux chart. There are no local symplectic invariants beyond dimension.

The symplectic neighborhood theorem says that a symplectomorphism between closed symplectic submanifolds which lifts to an isomorphism of their symplectic normal bundles extends to a symplectomorphism of neighborhoods.

Let $C_X\subset X$ and $C_Y\subset Y$ be the given copies of $C$. Their self-intersection numbers are the Euler classes of their oriented normal bundles, so square zero makes both normal bundles trivial. The neighborhood theorem identifies neighborhoods with $C\times D^2$. Remove their interiors and identify the boundary circle bundles by a map covering the chosen identification $C_X\cong C_Y$ and reversing the normal-circle orientation. On collars, the two forms have the modeland the radial coordinate can be reversed while the circle coordinate is reversed so that the forms glue. A collar application of Moser's trick removes any discrepancy. This proves that the symplectic fiber sum along a square-zero surface $X{\#}_{C}Y$ has a natural symplectic form.

The displayed relation is an ordinary product relation, sowhere the relation eliminates $b=ac{a}^{-1}$. The Gompf realization theorem constructs a closed symplectic four-manifold $M_{\Gamma }$ with this fundamental group. Concretely, its construction starts from a product of a sufficiently high-genus surface and a torus, represents the two surviving generators and the relations by loops, and crosses the relevant loops with circle factors to obtain square-zero tori. Symplectic sums with copies of the rational elliptic surface kill the unwanted generators and impose the relations: the complement of a regular elliptic fiber is simply connected, so the Seifert-van Kampen theorem gives exactly ${\pi }_1(M_{\Gamma })=\Gamma \cong F_2$.

Finally, ${\mathbb{CP}}^{48}$ is simply connected and has real dimension $96$. With the product symplectic form,is a closed symplectic manifold of real dimension $100$ and has fundamental group $\Gamma$.

Moser's trick states that if $M$ is compact and ${\omega }_t$, $0\le t\le 1$, is a smooth family of symplectic forms whose de Rham cohomology class is independent of $t$, then there is an isotopy ${\varphi }_t$ withSince $[{\dot{\omega }}_t]=0$, choose a smooth family of one-forms ${\sigma }_t$ with ${\dot{\omega }}_t=d{\sigma }_t$. Nondegeneracy of ${\omega }_t$ uniquely determines a vector field $X_t$ byCompactness makes its flow ${\varphi }_t$ exist for the whole interval. Cartan's magic formula and $d{\omega }_t=0$ givewhich proves the theorem.

Smooth degree-$d$ hypersurfaces form the complement of the discriminant in the projective space of degree-$d$ homogeneous polynomials. This complement is path connected, so $X$ and $X^{\prime }$ lie in a smooth one-parameter family. The Ehresmann fibration theorem identifies the fibers smoothly. Under such an identification, the restrictions of the Fubini-Study form form a family ${\omega }_t$ whose cohomology class is the fixed restricted hyperplane class. Moser's trick therefore gives the symplectic equivalence of smooth projective hypersurfaces.

It remains to construct the finite subgroup for one convenient hypersurface. On the Fermat hypersurfacethe group $({\mu }_d{)}^{n+1}$ acts by diagonal coordinate multiplication. It preserves both $X_F$ and the Fubini-Study form. The kernel of its projective action is the diagonal subgroup ${\mu }_d$, so the effective Fermat hypersurface diagonal symmetry group isConjugating this action by a symplectomorphism $X_F\to X$ gives the required subgroup of $\mathrm{Symp}(X,{\omega }_{FS}{|}_X)$.

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.