Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 146 (Symplectic Topology)

Use the compatible metrics $g_{\Sigma }(\cdot ,\cdot )={\omega }_{\Sigma }(\cdot ,j\cdot )$ and $g_X(\cdot ,\cdot )={\omega }_X(\cdot ,J\cdot )$. The energy of a smooth map isIt is a J-holomorphic curve when $du\circ j=J\circ du$. Splitting $du$ into its complex-linear and complex-antilinear parts gives the Energy identity for a J-holomorphic curveThe first term depends only on the homology class. It follows that a J-holomorphic map minimizes energy among all maps in its homology class.

One Monotonicity theorem for a J-holomorphic curve says that a nonconstant J-holomorphic curve through the center of a sufficiently small radius-$\rho$ ball, with boundary outside that ball, has area at least $c{\rho }^2$; in the standard complex ball one may take the sharp value $\pi {\rho }^2$. The Gromov non-squeezing theorem says

Write ${\mathbb{R}}^4=T^{\ast }{\mathbb{R}}^2$ with coordinates $(q,p)$ and form $d{q}_1\wedge d{p}_1+d{q}_2\wedge d{p}_2$. For $c>0$, the graphis a Lagrangian subspace. Points of $L_c\cap B(R)$ have $|q|\le R/\sqrt{1+c^2}$. Choose $c$ so large thatThen no two points of the $\epsilon$-neighborhood of $L_c\cap B(R)$ differ by a nonzero vector $(m,0)$ with $m\in {\mathbb{Z}}^2$, so quotienting $q$ modulo ${\mathbb{Z}}^2$ is injective there.

Relative to the Lagrangian splitting $L_c\oplus J{L}_c$, the map $(v,w)\mapsto (\lambda v,{\lambda }^{-1}w)$ is symplectic. It sends $B^4(r)$ into the indicated long thin neighborhood wheneverThese inequalities are compatible when $R>r^2/\epsilon$. Taking such an $R$ and then the quotient constructs the arbitrarily large symplectic balls in a cotangent cylinder:

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 $\Gamma$.

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 Poincare 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