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On the affine chart $[1:z_1:\cdots :z_n]$ of ${\mathbb{CP}}^n$, the normalized Fubini-Study form isIts integral over a projective line is $\pi$. The Darboux theorem says that every point of a symplectic $2n$-manifold has local coordinates in which the form is ${\sum }_{j}d{x}_j\wedge d{y}_j$.

To form the blowup ${\tilde{X}}_{\lambda }$, choose a symplectic embedding of the closed standard ball $B^{2n}(\lambda )$ centered at $p$, remove its interior, and collapse each characteristic Hopf circle of its boundary $S^{2n-1}$ to a point. The boundary becomes the exceptional divisor $E\cong {\mathbb{CP}}^{n-1}$, and the reduced form extends the old form outside the ball with every projective line in $E$ having area $\pi {\lambda }^2$. This is the symplectic blowup of size $\lambda$. In real dimension four its volume isThus, whenever two different sizes are allowed, different $\lambda$ give different total symplectic volumes and hence nonsymplectomorphic blowups.

The punctured area-$\pi$ sphere ${\mathbb{CP}}^1\setminus \{\infty \}$ is symplectomorphic to the open unit disc. Consequentlyfor every $r<1$. The bound is sharp. Given a hypothetical larger ball, choose a compatible almost complex structure agreeing with the pushed-forward standard structure on the ball. A standard J-holomorphic-curve result supplies a sphere in one ruling class through the ball center. Its total area is $\pi$, while monotonicity inside the ball requires at least $\pi r^2$. Hence $r\le 1$. Equivalently, the Gromov width of the monotone product of projective lines is

Fix $p\in X$. A linear change of coordinates first identifies ${\omega }_p$ with the standard symplectic form ${\omega }_0$. After shrinking to a star-shaped neighborhood, everyis nondegenerate. The Poincaré lemma gives $\omega -{\omega }_0=d\sigma$, with $\sigma (p)=0$. Define $X_t$ by ${\iota }_{X_t}{\omega }_t=-\sigma$ and let ${\varphi }_t$ be its local flow. Cartan's magic formula givesThus ${\varphi }_1^{\ast }\omega ={\omega }_0$, proving the Darboux theorem.

For a smooth function $H$ on a closed symplectic manifold, its Hamiltonian vector field is defined by ${\iota }_{X_H}\omega =-dH$. The same formula gives ${\mathcal{L}}_{X_H}\omega =-d^{2}H=0$, so the Hamiltonian flow preserves the symplectic form. To move one point to another in a connected $X$, join them by a path, cover the path by finitely many Darboux charts, and in each chart use a cutoff linear Hamiltonian to perform a small translation. Composing these compactly supported Hamiltonian diffeomorphisms proves that symplectomorphisms act transitively on each connected component.

In ${\mathbb{R}}^2$, every embedded curve is a Lagrangian submanifold. Let $L_1,L_2$ be circles enclosing different Euclidean areas. A plane symplectomorphism preserves area and carries the bounded complementary component of one circle to that of its image, so no symplectomorphism maps $L_1$ to $L_2$.

The same phenomenon exists in every ${\mathbb{R}}^{2n}$. Withtake the product toriThe Liouville class of a Lagrangian submanifold has respective period vectors $\pi (1,\dots ,1)$ and $2\pi (1,\dots ,1)$ on these tori. Every symplectomorphism of ${\mathbb{R}}^{2n}$ preserves the Liouville class up to the induced integral change of basis on $H_1(T^n)$, because its pullback changes $\lambda$ only by an exact form. An integral automorphism sends a primitive vector to a primitive vector and therefore cannot send the first period vector to the second. Hence