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For every \( \theta \in {\mathbb{R}}^{m} \) the function \( H = {H}_{\theta } = \langle \theta ,\mu \rangle : M \rightarrow \mathbb{R} \) is a Morse–Bott function with even-dimensional critical manifolds of even index. Moreover, the critical set\n\n\[ \n\operatorname{Crit}\left( {H}_{\theta }\right) = \mathop{\bigcap }... | Proof: Assume first that \( \theta \) has rationally independent components so that the vectors \( {t\theta } + k \) for \( t \in \mathbb{R} \) and \( k \in {\mathbb{Z}}^{m} \) form a dense set in \( {\mathbb{R}}^{m} \) . Then every critical point \( x \) of \( H = \langle \theta ,\mu \rangle \) is fixed under the acti... | Yes |
Theorem 5.6.1 (Duistermaat-Heckman)\n\nConsider a circle action on a closed manifold \( \\left( {M,\\omega }\\right) \) that is generated by a Morse function \( H : M \\rightarrow \\mathbb{R} \) . Then\n\n\[ \n{\\int }_{M}{e}^{-\\hslash H}\\frac{{\\omega }^{n}}{n!} = \\mathop{\\sum }\\limits_{p}\\frac{{e}^{-\\hslash H\... | Proof: See page 245. | No |
Lemma 5.6.3 (Localization) Assume that the action has isolated fixed points and\n\n\[ \tau = {\tau }_{2n} + \hslash {\tau }_{{2n} - 2} + \cdots + {\hslash }^{n}{\tau }_{0} \]\n\nis a \( {d}_{\hslash } \) -closed \( {2n} \) -form such that \( {\tau }_{0} \) vanishes on the fixed points of the action. Then \( \tau \) is ... | Proof: Assume first that the action has no fixed points. Then the vector field \( X \) has no zeros and so there exists a 1 -form \( \alpha \in {\Omega }^{1}\left( M\right) \) such that\n\n\[ \iota \left( X\right) \alpha = 1,\;\iota \left( X\right) {d\alpha } = 0. \]\n\nTo see this just choose any 1 -form with \( \alph... | Yes |
Lemma 5.6.4 Assume that the circle action is Hamiltonian and the critical points of \( H \) are all nondegenerate. Then for every fixed point \( p \) there exists an equivariant differential form\n\n\[ \n{\tau }_{p} = {\tau }_{p,{2n}} + \hslash {\tau }_{p,{2n} - 2} + \cdots + {\hslash }^{n}{\tau }_{p,0}\; \in \;{\Omega... | Proof: Consider the splitting\n\n\[ \n{T}_{p}M = {L}_{1} \oplus \cdots \oplus {L}_{n} \n\]\n\ninto complex subspaces (with respect to an invariant complex structure) such that \( {S}^{1} \) acts on \( {L}_{j} \) by multiplication with \( {\theta }^{-{k}_{j}} \) . Now let\n\n\[ \n{\phi }_{j} : {L}_{j} \rightarrow {S}^{2... | Yes |
Example 5.7.1 The most basic example is given by the action of \( \mathrm{G} = {S}^{1} \) on \( {S}^{2} = \mathbb{C} \cup \{ \infty \} \) by rotation about the origin. In this case \( {\mathrm{G}}^{\mathrm{c}} = {\mathbb{C}}^{ * } \) has precisely two fixed points at \( 0,\infty \), all other points lying on a single \... | If we take a moment map \( \mu : {S}^{2} \rightarrow \mathbb{R} \) in which \( {\mu }^{-1}\left( 0\right) \) is a circle, then \( {M}^{\mathrm{s}} = {M}^{\mathrm{{ss}}} = {M}^{\mathrm{{ps}}} = \mathbb{C} \smallsetminus \{ 0\} \), while \( {M}^{\mathrm{{us}}} \) consists of the two fixed points. However, if \( \mu \) is... | No |
Theorem 5.7.2 For every \( p \in M \) the following holds.\n\n(i) \( p \) is \( \mu \) -semistable if and only if \( {w}_{\mu }\left( {p,\xi }\right) \geq 0 \) for all \( \xi \in \Lambda \) .\n\n(ii) \( p \) is \( \mu \) -polystable if and only if \( {w}_{\mu }\left( {p,\xi }\right) \geq 0 \) for all \( \xi \in \Lambda... | Proof: See Mumford-Fogarty-Kirwan [497] in the algebraic geometric setting and [262] for Hamiltonian group actions on general closed Kähler manifolds. | No |
Lemma 6.1.3 Let \( \pi : M \rightarrow B \) be a locally trivial fibration with connected base and \( \omega \in {\Omega }^{2}\left( M\right) \) be a symplectic form such that the fibres are all symplectic submanifolds of \( M \) . Then \( \pi : M \rightarrow B \) admits the structure of a symplectic fibration which is... | Proof: This is an exercise with hints. First use Stokes' theorem to prove that the symplectic forms\n\n\[{\sigma }_{b} = {\iota }_{b}^{ * }\omega \in {\Omega }^{2}\left( {F}_{b}\right)\]\n\nall represent the same cohomology class in \( {H}^{2}\left( F\right) \) under the local trivializations \( {\phi }_{\alpha }\left(... | No |
Consider the Hopf surface\n\n\[ M = {S}^{3} \times {S}^{1} = \left( {{\mathbb{C}}^{2}\smallsetminus \{ 0\} }\right) /\mathbb{Z} \]\n\nwhere the generator of \( \mathbb{Z} \) acts on \( {\mathbb{C}}^{2} \smallsetminus \{ 0\} \) by multiplication by 2 . This manifold does not admit a symplectic structure because \( {H}^{... | However, it is easy to see that the projection\n\n\[ \pi : M \rightarrow {\mathbb{{CP}}}^{1} \]\n\ndefined by \( \pi \left( z\right) = \left\lbrack {{z}_{0} : {z}_{1}}\right\rbrack \) for \( z = \left( {{z}_{0},{z}_{1}}\right) \in {\mathbb{C}}^{2} \smallsetminus \{ 0\} \), is a symplectic fibration with fibre \( {\math... | No |
Lemma 6.1.7 Let \( \pi : M \rightarrow B \) be a symplectic fibration with fibres symplecto-morphic to \( \left( {F,\sigma }\right) \) . Assume that the first Chern class \( {c}_{1}\left( {TF}\right) \in {H}^{2}\left( {F;\mathbb{R}}\right) \) is a nonzero multiple of the class \( \left\lbrack \sigma \right\rbrack \) . ... | Proof: By assumption, \( {c}_{1}\left( {TF}\right) = \lambda \left\lbrack \sigma \right\rbrack \) for some constant \( \lambda \neq 0 \) and hence \( {c}_{1}\left( {T{F}_{b}}\right) = \lambda \left\lbrack {\sigma }_{b}\right\rbrack \) for every \( b \in B \) . Now consider the vector bundle\n\n\[ \n\text{Vert} \subset ... | Yes |
Corollary 6.1.8 Let \( \\left( {F,\\sigma }\\right) \) be a compact symplectic manifold whose first Chern class \( {c}_{1}\\left( {TF}\\right) \\in {H}^{2}\\left( {F;\\mathbb{R}}\\right) \) is a nonzero multiple of the class \( \\left\\lbrack \\sigma \\right\\rbrack \) . Then the total space \( M \) of any symplectic f... | Proof: Theorem 6.1.4 and Lemma 6.1.7. | No |
Theorem 6.2.2 Let \( F \) be a compact oriented 2-manifold of genus \( g\left( F\right) \neq 1 \) . Then the total space of any oriented fibration \( \pi : M \rightarrow B \) with fibre \( F \) and compact symplectic base \( B \) admits a symplectic form that is compatible with the oriented fibration. | Proof: We first claim that \( \pi \) admits the structure of a symplectic fibration. The proof is based on the observation that the structure group of a principal G-bundle \( P \rightarrow B \) can be reduced to the subgroup \( \mathrm{H} \subset \mathrm{G} \) if and only if the associated \( \mathrm{G}/\mathrm{H} \) -... | Yes |
Lemma 6.2.3 For every closed oriented 2-manifold \( B \) there are exactly two orientable \( {S}^{2} \) -bundles with base \( B \), namely the product \( B \times {S}^{2} \) and the nontrivial bundle \( {X}_{B} \) . | Proof: As in Section 2.6, one can prove that every \( {S}^{2} \) -bundle over a compact connected Riemann surface with boundary admits a trivialization. Hence, split \( B = {B}_{1}{ \cup }_{C}{B}_{2} \) and trivialize the bundle over each half. Thus\n\n\[ X = \left( {{B}_{1} \times {S}^{2}}\right) \cup \left( {{B}_{2} ... | Yes |
Theorem 6.2.5 Let \( a \mathrel{\text{:=}} {\mu }_{ + }{b}_{ + } + {\mu }_{ - }{b}_{ - } \in {H}^{2}\left( {{X}_{B};\mathbb{R}}\right) \), where \( {b}_{ \pm } \in {H}^{2}\left( {{X}_{B};\mathbb{R}}\right) \) are defined by (6.2.1) and \( {\mu }_{ \pm } \in \mathbb{R} \) .\n\n(i) If \( B = {S}^{2} \) then there is a sy... | Proof: See \( \left\lbrack {{449},{454}}\right\rbrack \) and the exercises below. | No |
Theorem 6.3.3 (Weinstein) Let \( \mathrm{G} \rightarrow \operatorname{Symp}\left( {F,\sigma }\right) : g \mapsto {\psi }_{g} \) be a Hamiltonian group action. Then every connection on a principal G-bundle \( P \rightarrow B \) gives rise to a closed 2-form \( \tau \) on the associated fibration \( P{ \times }_{\mathrm{... | Proof 1: Denote by \( \mathfrak{g} \rightarrow \operatorname{Vect}\left( F\right) : \xi \mapsto {X}_{\xi } \) the infinitesimal action of the Lie algebra \( \mathfrak{g} \mathrel{\text{:=}} \operatorname{Lie}\left( \mathrm{G}\right) \) and let \( \mu : F \rightarrow {\mathfrak{g}}^{ * } \) be a moment map for the actio... | Yes |
Lemma 6.3.5 Assume that \( \tau \in {\Omega }^{2}\left( M\right) \) satisfies \( {\iota }_{b}^{ * }\tau = {\sigma }_{b} \) for every \( b \in B \) . Then the connection \( {\Gamma }_{\tau } \) is symplectic if and only if \( \tau \) is vertically closed in the sense that \( {d\tau }\left( {{\eta }_{1},{\eta }_{2}, \cdo... | Proof 1: Given a vector field \( v : B \rightarrow {TB} \) on the base \( B \) denote by\n\n\[ \n{v}^{\sharp } : M \rightarrow {TM} \n\]\n\nits horizontal lift. Then the flow of \( {v}^{\sharp } \) preserves the fibres of \( \pi \) . The connection \( {\Gamma }_{\tau } \) is symplectic if and only if the flow of every ... | Yes |
Theorem 6.4.1 Let \( \pi : M \rightarrow B \) be a symplectic fibration and \( \Gamma \) be a symplectic connection on \( M \) . Then the following are equivalent.\n\n(i) There exists a closed connection 2 -form \( \tau \in {\Omega }^{2}\left( M\right) \) such that \( \Gamma = {\Gamma }_{\tau } \) .\n\n(ii) The holonom... | Proof: See pages 273 and 280. | No |
Corollary 6.4.3 Let \( \\left( {F,\\sigma }\\right) \) be a compact simply connected symplectic manifold. Then the total space \( M \) of any symplectic fibration \( \\pi : M \\rightarrow B \) with fibre \( \\left( {F,\\sigma }\\right) \) and compact symplectic base \( B \) carries a compatible symplectic structure. | Proof: Condition (ii) in Theorem 6.4.1 is automatically satisfied whenever the fibre \( F \) is simply connected. Hence Theorem 6.4.1 shows that there exists a closed 2 -form \( \\tau \\in {\\Omega }^{2}\\left( M\\right) \) such that \( {\\iota }_{b}^{ * }\\tau = {\\sigma }_{b} \) for every \( b \\in B \) . Let \( \\be... | Yes |
Lemma 6.4.8 (Curvature identity [292]) Let \( \tau \in \mathcal{T}\left( {M,\sigma }\right) \) be a connection 2-form. Then the curvature of the connection \( \Gamma = {\Gamma }_{\tau } \) satisfies\n\n\[ \iota \left( {\left\lbrack {v}_{1}^{\sharp },{v}_{2}^{\sharp }\right\rbrack }^{\text{Vert }}\right) \tau \overset{\... | Proof: The connection 2-form \( \tau \) satisfies\n\n\[ {d\tau }\left( {{v}_{1}^{\sharp },{v}_{2}^{\sharp }, Y}\right) = \tau \left( {\left\lbrack {{v}_{1}^{\sharp },{v}_{2}^{\sharp }}\right\rbrack, Y}\right) + \tau \left( {\left\lbrack {{v}_{2}^{\sharp }, Y}\right\rbrack ,{v}_{1}^{\sharp }}\right) + \tau \left( {\left... | Yes |
Theorem 6.5.3 Let \( \pi : M \rightarrow B \) be a symplectic fibration whose restriction to every loop in \( B \) admits a symplectic trivialization. The following are equivalent.\n\n(i) \( \pi : M \rightarrow B \) is a Hamiltonian fibration.\n\n(ii) The structure group reduces to \( \operatorname{Ham}\left( {F,\sigma... | Proof: See page 285. | No |
Lemma 7.1.1 Let \( h \in {H}^{2}\left( {{\mathbb{{CP}}}^{n - 1};\mathbb{Z}}\right) \) be the positive generator. Then the first Chern class of the tautological line bundle \( L = {\widetilde{\mathbb{C}}}^{n} \rightarrow {\mathbb{{CP}}}^{n - 1} \) is \( {c}_{1}\left( L\right) = - h \) . In particular, if \( n = 2 \) the... | Proof: This is Exercise 2.7.8. Here is another approach to this calculation, based on Theorem 2.7.5. Because the tautological line bundle \( L \rightarrow {\mathbb{{CP}}}^{n - 1} \) restricts to the tautological bundle on \( {\mathbb{{CP}}}^{1} \subset {\mathbb{{CP}}}^{n - 1} \), it suffices to prove that the first Che... | No |
Lemma 7.1.3 Define\n\n\[ \widetilde{M} \mathrel{\text{:=}} \left( {M \smallsetminus \left\{ {p}_{0}\right\} }\right) \cup Z,\;{\pi }_{M} : \widetilde{M} \rightarrow M \]\n\nby \( {\left. {\pi }_{M}\right| }_{M \smallsetminus \left\{ {p}_{0}\right\} } \mathrel{\text{:=}} \mathrm{{id}} \) and \( {\pi }_{M}\left( \widetil... | Proof of Lemma 7.1.3: We construct an atlas on \( \widetilde{M} \) with holomorphic transition maps. Associated to every holomorphic coordinate chart \( \phi : U \rightarrow {\mathbb{C}}^{n} \) on \( M \), defined on an open set \( U \subset M \smallsetminus \left\{ {p}_{0}\right\} \), is a coordinate chart on \( \wide... | No |
Example 7.1.5 Consider the manifold\n\n\\[ X \\mathrel{\\text{:=}} {\\mathbb{{CP}}}^{2}\\# {\\overline{\\mathbb{{CP}}}}^{2} \\]\n\nobtained by blowing up \\( {\\mathbb{{CP}}}^{2} \\) at the point \\( {p}_{0} \\mathrel{\\text{:=}} \\left\\lbrack {1 : 0 : 0}\\right\\rbrack \\) . The family of lines in \\( {\\mathrm{{CP}}... | Observe that \\( X \\) can be embedded in \\( {\\mathbb{{CP}}}^{1} \\times {\\mathbb{{CP}}}^{2} \\) as the submanifold\n\n\\[ X = \\left\\{ {\\left( {\\left\\lbrack {{w}_{1} : {w}_{2}}\\right\\rbrack ,\\left\\lbrack {{z}_{0} : {z}_{1} : {z}_{2}}\\right\\rbrack }\\right) \\mid {w}_{1}{z}_{2} = {w}_{2}{z}_{1}}\\right\\} ... | Yes |
Example 7.1.7 (Elliptic surfaces) A complex surface \( M \) is said to be elliptic if there is a holomorphic map \( f : M \rightarrow {\mathbb{{CP}}}^{1} \) whose generic fibre \( {f}^{-1}\left( z\right) \) is an elliptic curve, i.e. is biholomorphic to the 2-torus equipped with a linear complex structure. In general, ... | Let \( V \) be the manifold obtained by blowing up \( {\mathbb{{CP}}}^{2} \) at nine distinct points. The complex structure on \( V \) varies depending on which set \( X \) of nine points one blows up. One case of particular interest is when the nine points are the points of intersection of two transverse nonsingular c... | Yes |
Example 7.1.8 (Lefschetz pencils) More generally all projective Kähler surfaces \( X \) support a Lefschetz pencil, which is a one-parameter family of complex curves, finitely many of them singular, that all go through the same finite number of points and are otherwise distinct. If \( X \) is blown up at these points, ... | (Cf. Theorem 7.4.12 and Example 7.4.13.) | No |
Example 7.1.9 (Resolving singularities) (i) Consider the variety \( V \subset {\mathbb{C}}^{2} \) defined by the equation \( {z}_{1}{z}_{2} = 0 \) . This has a singularity at 0 where the two lines \( {z}_{1} = 0 \) and \( {z}_{2} = 0 \) cross. But, because these lines represent different points in the exceptional divis... | \[ \widetilde{V} \mathrel{\text{:=}} \overline{{\pi }^{-1}\left( {V\smallsetminus \{ 0\} }\right) } \subset {\widetilde{\mathbb{C}}}^{2}. \] | Yes |
Lemma 7.1.11 (i) Let \( f : \left( {0,\infty }\right) \rightarrow \left( {0,\infty }\right) \) be a smooth function with positive derivative and define \( F : {\mathbb{C}}^{n} \smallsetminus \{ 0\} \rightarrow {\mathbb{C}}^{n} \) by\n\n\[ F\left( z\right) \mathrel{\text{:=}} f\left( \left| z\right| \right) \frac{z}{\le... | Proof: We prove that \( {F}^{ * }{\omega }_{0} \) is of type \( \left( {1,1}\right) \) . To see this, note that\n\n\[ {F}^{ * }d{z}_{j} = \frac{f\left( \left| z\right| \right) }{\left| z\right| }d{z}_{j} + \frac{\left| z\right| {f}^{\prime }\left( \left| z\right| \right) - f\left( \left| z\right| \right) }{2{\left| z\r... | Yes |
The Kähler forms \( {\widetilde{\omega }}_{\lambda ,\varepsilon } \in {\Omega }^{2}\left( {\widetilde{\mathbb{C}}}^{n}\right) \) in Definition 7.1.12 depend smoothly on \( \lambda > 0 \) and \( 0 < \varepsilon < 1 \), and they satisfy the equations \[ {\left. {\widetilde{\omega }}_{\lambda ,\varepsilon }\right| }_{L\le... | Proof: By definition the 2 -forms \( {\widetilde{\omega }}_{\lambda ,\varepsilon } \) depend smoothly on \( \lambda \) and \( \varepsilon \), and they satisfy equation (7.1.11) by definition of \( {F}_{\lambda ,\varepsilon } \) . To prove (7.1.12), recall from Exercise 4.3.4 that \( {\rho }_{\mathrm{{FS}}} = d{\alpha }... | No |
Example 7.1.16 By Example 7.1.5, the manifold \( X \) obtained by blowing up one point of \( \left( {{\mathbb{{CP}}}^{2},{\omega }_{\mathrm{{FS}}}}\right) \) is the nontrivial \( {S}^{2} \) -bundle over \( {S}^{2} \) . Thus, we now have two ways of putting a symplectic form on \( X \), one by symplectic blowup and the ... | For example, one can take \( \psi \) to be the composite of the inclusion \( B\left( \lambda \right) \hookrightarrow \operatorname{int}B\left( 1\right) \) with the symplectomorphism \( \iota : \left( {\operatorname{int}B\left( 1\right) ,{\omega }_{0}}\right) \rightarrow \left( {{\mathbb{{CP}}}^{2} \smallsetminus {\math... | Yes |
Theorem 7.1.21 (Symplectic blowup) Assume \( \left( {M,\omega, J}\right) \) is normalized at \( {p}_{0} \) with \( \delta ,{\psi }_{0} \) as in Definition 7.1.17 and let \( \left( {\widetilde{M},\widetilde{J}}\right) \) be the almost complex blowup of \( M \) at \( {p}_{0} \) in Definition 7.1.19. For \( \lambda > 0 \)... | Here \( \widetilde{\psi } : L\left( r\right) \rightarrow \widetilde{M} \) is the unique embedding that satisfies \( {\pi }_{M} \circ \widetilde{\psi } = \psi \circ \pi \) . The symplectic forms \( {\widetilde{\omega }}_{\psi ,\lambda ,\varepsilon } \) have the following properties. (i) Let \( {\widetilde{\omega }}_{\la... | Yes |
Proposition 7.1.22 (Normalization of ball embeddings) Let \( \left( {M,\omega }\right) \) be a connected symplectic manifold without boundary.\n\n(i) Let \( r > 0 \) and let \( {\psi }_{0},{\psi }_{1} : B\left( r\right) \rightarrow M \) be symplectic embeddings. Then there exists a Hamiltonian symplectomorphism \( \phi... | Proof: We prove (i). Choose a smooth path \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow M \) with endpoints \( \gamma \left( 0\right) = {\psi }_{0}\left( 0\right) \) and \( \gamma \left( 1\right) = {\psi }_{1}\left( 0\right) \) and let \( {\left\{ {\chi }_{t}\right\} }_{0 \leq t \leq 1} \) be a Hamiltonian is... | Yes |
Theorem 7.1.23 (Uniqueness of blowup) Assume \( \left( {M,\omega, J}\right) \) is normalized at \( {p}_{0} \) by \( {\psi }_{0} : B\left( \delta \right) \rightarrow {U}_{0} \), and let \( {\pi }_{M} : \widetilde{M} \rightarrow M \) be the almost complex blowup in Definition 7.1.19. For \( i = 1,2 \) let \( {\psi }_{i} ... | Proof: The symplectic forms \( {\widetilde{\omega }}_{{\psi }_{1},{\lambda }_{1},{\varepsilon }_{1}} \) and \( {\widetilde{\omega }}_{{\psi }_{2},{\lambda }_{2},{\varepsilon }_{2}} \) agree whenever\n\n\[{\lambda }_{1} = {\lambda }_{2} = : \lambda ,\;{\varepsilon }_{1} = {\varepsilon }_{2} = : \varepsilon ,\;\lambda + ... | Yes |
Corollary 7.1.24 Let \( \lambda > 0 \) . If there exists a symplectic embedding \( B\left( \lambda \right) \hookrightarrow M \) , then the de Rham cohomology class \( \widetilde{a} - \pi {\lambda }^{2}e \) on \( \widetilde{M} \) has a symplectic representative with first Chern class \( \widetilde{c} - \left( {n - 1}\ri... | Proof: Every symplectic embedding \( \psi : B\left( \lambda \right) \rightarrow M \) extends to \( B\left( r\right) \) for some constant \( r > \lambda \) by part (i) of Theorem 3.3.1, and can be normalized near the origin by part (i) of Proposition 7.1.22. Hence the assertion follows from part (v) of Theorem 7.1.21. | Yes |
Lemma 7.1.25 If \( M \) is simply connected then the inclusion\n\n\[ \n{\operatorname{Emb}}_{{p}_{0}}\left( {B\left( \lambda \right), M}\right) \hookrightarrow \operatorname{Emb}\left( {B\left( \lambda \right), M}\right) \n\]\n\ninduces an isomorphism on the space of path-components. | Proof 1: Let \( {\psi }_{0},{\psi }_{1} \in {\operatorname{Emb}}_{{p}_{0}}\left( {B\left( \lambda \right), M}\right) \) and let \( \left\lbrack {0,1}\right\rbrack \rightarrow \operatorname{Emb}\left( {B\left( \lambda \right), M}\right) : t \mapsto {\psi }_{t} \) be a smooth family of symplectic embeddings connecting \(... | Yes |
Corollary 7.1.26 Assume \( M \) is simply connected and let \( {\psi }_{0},{\psi }_{1} : B\left( \lambda \right) \rightarrow M \) be normalized symplectic embeddings. If \( {\psi }_{0} \) and \( {\psi }_{1} \) are isotopic through symplectic embeddings of \( B\left( \lambda \right) \), then the symplectic forms \( {\wi... | Proof: By Lemma 7.1.25 the embeddings \( {\psi }_{0},{\psi }_{1} \) are based symplectically isotopic. Hence the assertion follows from part (iii) of Theorem 7.1.23. | Yes |
Lemma 7.2.2 Let \( \left( {{M}_{1},{\omega }_{1}}\right) \) and \( \left( {{M}_{2},{\omega }_{2}}\right) \) be two compact symplectic manifolds of dimension \( {2n} \neq 2,6 \) . Then the connected sum \( M \mathrel{\text{:=}} {M}_{1}\# {M}_{2} \) does not carry a symplectic form which on each half \( {M}_{i} \smallset... | Proof: Assume by contradiction that such a form \( \omega \) exists and let \( J \) be an \( \omega \) -tame almost complex structure on \( M \) . Then its restriction \( {J}_{i} \) to \( {M}_{i} \smallsetminus {B}_{i} \) is homotopic to any \( {\omega }_{i} -tame almost complex structure and hence to one which extends... | Yes |
Theorem 7.3.1 (Gromov) Let \( M \) be an open 2n-dimensional manifold. Let \( \tau \in {\Omega }^{2}\left( M\right) \) be a nondegenerate 2-form and let \( a \in {H}^{2}\left( {M;\mathbb{R}}\right) \) . (i) There exists a smooth family of nondegenerate 2 -forms \( {\tau }_{t} \) on \( M \) such that \( {\tau }_{0} = \t... | Proof: We shall prove the theorem in the case \( a = 0 \) . The strategy is to exhaust \( M \) by the sublevel sets \[ {M}^{c} \mathrel{\text{:=}} \{ q \in M \mid f\left( q\right) \leq c\} \] of a Morse function \( f : M \rightarrow \mathbb{R} \) that is proper and bounded below, and does not have any critical points o... | Yes |
Lemma 7.3.2 Let \( \Lambda \subset {\mathbb{R}}^{N} \) be a compact set and let \( 1 \leq m < {2n} \) . Assume that\n\n\[ \Lambda \rightarrow {\Omega }^{2}\left( {I}^{2n}\right) : \lambda \mapsto {\tau }_{\lambda } \]\n\nis a smooth family of nondegenerate 2-forms such that \( {\tau }_{\lambda } \) is exact in a neighb... | Proof: The proof is by induction over \( m \) and involves the telescope construction. The first step is the case \( m = 1 \) and we shall first assume that \( \Lambda \) is a single point. Thus we are given a 2-form\n\n\[ \tau \in {\Omega }^{2}\left( {I \times {I}^{{2n} - 1}}\right) \]\n\nwhose restriction to a neighb... | Yes |
Example 7.3.3 Suppose that \( \left( {{M}^{2n}, J, a}\right) \) is a closed, connected, almost complex manifold with \( a \in {H}^{2}\left( {M;\mathbb{R}}\right) \) such that \( {a}^{n} > 0 \), and let \( {p}_{0} \) be any point in \( M \) . In dimensions greater than four it is unknown whether such \( M \) must suppor... | However, Gromov constructs a symplectic form \( \omega \) on \( M \smallsetminus \left\{ {p}_{0}\right\} \) in the class \( a \) that is tamed by some almost complex structure homotopic to \( J \) . One might hope to construct such a form that is controlled near \( {p}_{0} \) in the sense that near \( {p}_{0} \) it equ... | Yes |
Theorem 7.4.1 (Donaldson) Let \( \left( {M,\omega }\right) \) be a closed symplectic 2n-manifold and suppose that the cohomology class \( \left\lbrack \omega \right\rbrack \in {H}^{2}\left( {M;\mathbb{R}}\right) \) admits an integral lift. Then, for every sufficiently large integer \( k \) and every integral lift \( a ... | Proof: See Donaldson [146] and also Auroux [36]. | No |
Lemma 7.4.2 Let \( L \rightarrow M \) be a complex line bundle with Chern class \( {c}_{1}\left( L\right) = a \) . Assume \( L \) carries a Hermitian structure and let \( D = \partial + \bar{\partial } \) be a Hermitian connection. If a section \( s : M \rightarrow L \) satisfies (7.4.1) on its zero set then \( s \) is... | Proof: Assume that \( L \) carries a Hermitian structure and let \( D \) be a Hermitian connection. Choose local coordinates \( z = \left( {{x}_{1},\ldots ,{x}_{n},{y}_{1},\ldots ,{y}_{n}}\right) \) on \( M \) such that the almost complex structure \( J \) is standard at \( z = 0 \) and the section \( s : U \rightarrow... | Yes |
Theorem 7.4.12 (Donaldson) Let \( \left( {M,\omega }\right) \) be a compact symplectic 4-manifold and suppose that the cohomology class \( \left\lbrack \omega \right\rbrack \) admits an integral lift. Denote\n\n\[ m \mathrel{\text{:=}} {\int }_{M}\omega \land \omega \]\n\nThen, for every sufficiently large integer \( k... | Proof: See Donaldson [148]. | No |
Lemma 8.1.1 The Floquet multipliers of \( p\left( t\right) \) are independent of the choice of the Poincaré section. | Proof: Let \( {\sum }^{\prime } \mathrel{\text{:=}} \left\{ {p \in \Omega \mid {G}^{\prime }\left( p\right) = H\left( p\right) = 0}\right\} \) be another Poincaré section with \( {G}^{\prime }\left( {p}_{0}^{\prime }\right) = 0 \), where \( {p}_{0}^{\prime } \mathrel{\text{:=}} p\left( {t}_{0}^{\prime }\right) \) . Let... | Yes |
Lemma 8.1.2 The Poincaré section \( \sum \cap U \) is a symplectic submanifold of \( M \) and the Poincaré section map \( \psi : \sum \cap U \rightarrow \sum \) is a symplectomorphism. | Proof: The hypersurface \( \sum \) is of dimension \( {2n} - 2 \) and the tangent space at \( p \) is\n\n\[ \n{T}_{p}\sum = \left\{ {v \in {T}_{p}M \mid {dG}\left( p\right) v = {dH}\left( p\right) v = 0}\right\} .\n\]\n\nThe condition \( \{ G, H\} = \omega \left( {{X}_{G},{X}_{H}}\right) \neq 0 \) shows that the 2-dime... | Yes |
Theorem 8.2.1 (Poincaré-Birkhoff) Let \( \psi \) be an area-preserving homeomorphism of the annulus \( A \mathrel{\text{:=}} \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} \mid a \leq y \leq b}\right\} \) satisfying (8.2.1),(8.2.2), and (8.2.3). Then \( \psi \) has at least two geometrically distinct fixed points. | Proof: See page 347 for the existence of one fixed point. | No |
Corollary 8.2.3 Let \( \psi \) be an area-preserving homeomorphism of the annulus satisfying (8.2.1) and (8.2.2), and suppose that\n\n\[ m = \mathop{\max }\limits_{x}\left( {f\left( {x, a}\right) - x}\right) < \mathop{\min }\limits_{x}\left( {f\left( {x, b}\right) - x}\right) = M. \]\n\nThen \( \psi \) has infinitely m... | Proof: Denote by \( {f}^{q}\left( {x, y}\right) \) and \( {g}^{q}\left( {x, y}\right) \) the first and second components of the \( q \) th iterate of \( \psi \) . Then\n\n\[ {f}^{q}\left( {x, a}\right) - x = \mathop{\sum }\limits_{{j = 0}}^{{q - 1}}\left( {{f}^{j + 1}\left( {x, a}\right) - {f}^{j}\left( {x, a}\right) }... | Yes |
Lemma 8.3.1 Under the above assumptions there exists a function \( h : U \rightarrow \mathbb{R} \) such that, for all \( \left( {{x}_{0},{y}_{0}}\right) \in \Omega \) and \( \left( {{x}_{1},{y}_{1}}\right) \in {\mathbb{R}}^{2} \) with \( \left( {{x}_{0},{x}_{1}}\right) \in U \), we have\n\n\[ \left( {{x}_{1},{y}_{1}}\r... | Proof: By assumption the map\n\n\[ \Omega \rightarrow U : \left( {{x}_{0},{y}_{0}}\right) \mapsto \left( {{x}_{0}, f\left( {{x}_{0},{y}_{0}}\right) }\right) \]\n\nhas a smooth inverse \( U \rightarrow \Omega : \left( {{x}_{0},{x}_{1}}\right) \mapsto \left( {{x}_{0}, - u\left( {{x}_{0},{x}_{1}}\right) }\right) \), where... | Yes |
Lemma 8.3.2\n\n\\[ h\\left( {{x}_{0} + 1,{x}_{1} + 1}}\\right) = h\\left( {{x}_{0},{x}_{1}}\\right) .\n\\] | Proof: First note that\n\n\\[ u\\left( {{x}_{0} + 1,{x}_{1} + 1}}\\right) = u\\left( {{x}_{0},{x}_{1}}\\right) ,\\;v\\left( {{x}_{0} + 1,{x}_{1} + 1}}\\right) = v\\left( {{x}_{0},{x}_{1}}\\right) .\n\\]\n\nHence\n\n\\[ \\frac{\\partial }{\\partial {x}_{i}}\\left( {h\\left( {{x}_{0} + 1,{x}_{1} + 1}}\\right) - h\\left( ... | Yes |
Lemma 9.1.1 Under the above assumptions there exists a function \( S : U \rightarrow \mathbb{R} \) such that, for all \( \left( {{x}_{0},{y}_{0}}\right) \in \Omega \) and \( \left( {{x}_{1},{y}_{1}}\right) \in {\mathbb{R}}^{2n} \) with \( \left( {{x}_{0},{x}_{1}}\right) \in U \), we have\n\n\[ \left( {{x}_{1},{y}_{1}}\... | Proof: By assumption, the map \( \Omega \rightarrow U : \left( {{x}_{0},{y}_{0}}\right) \rightarrow \left( {{x}_{0}, u\left( {{x}_{0},{y}_{0}}\right) }\right) \) has a smooth inverse \( U \rightarrow \Omega : \left( {{x}_{0},{x}_{1}}\right) \mapsto \left( {{x}_{0}, - f\left( {{x}_{0},{x}_{1}}\right) }\right) \), where ... | Yes |
A linear symplectomorphism of the form\n\n\[ \Psi = \left( \begin{array}{ll} A & B \\ C & D \end{array}\right) \]\n\nadmits a generating function \( S = S\left( {{x}_{0},{x}_{1}}\right) \) if and only if \( \det \left( B\right) \neq 0 \) . | A generating function is given by\n\n\[ S\left( {{x}_{0},{x}_{1}}\right) \mathrel{\text{:=}} \frac{1}{2}\left\langle {{x}_{0},{B}^{-1}A{x}_{0}}\right\rangle - \left\langle {{x}_{0},{B}^{-1}{x}_{1}}\right\rangle + \frac{1}{2}\left\langle {{x}_{1}, D{B}^{-1}{x}_{1}}\right\rangle \]\n\nfor \( {x}_{0},{x}_{1} \in {\mathbb{... | Yes |
Lemma 9.1.4 The function \( {S}_{H} \) is a generating function for \( \psi = {\phi }_{H}^{{t}_{1},{t}_{0}} \) . | Proof: Choose \( \left( {{x}_{0},{x}_{1}}\right) \in U \) and \( \left( {{\xi }_{0},{\xi }_{1}}\right) \in {\mathbb{R}}^{n} \times {\mathbb{R}}^{n} \) . For small \( s \in \mathbb{R} \) denote by \( {z}_{s}\left( t\right) = \left( {{x}_{s}\left( t\right) ,{y}_{s}\left( t\right) }\right) \) the unique solution of Hamilt... | Yes |
Lemma 9.1.6 If \( x : \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack \rightarrow \Omega \) satisfies \( \dot{x} = f\left( x\right) \) and \( \xi : \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack \rightarrow \Omega \) is any function such that \( \xi \left( {t}_{0}\right) = x\left( {t}_{0}\right) = {x}_{0} \) and \( \xi \left(... | Proof: Since the matrix \( {\partial }^{2}L/\partial {v}^{2} \) is positive definite there is an inequality\n\n\[L\left( {\xi, f\left( \xi \right) }\right) + \left\langle {{\partial }_{v}L\left( {\xi, f\left( \xi \right) }\right), v - f\left( \xi \right) }\right\rangle \leq L\left( {\xi, v}\right)\]\n\n(9.1.11)\n\nfor ... | Yes |
Lemma 9.2.1 If \( \psi \) satisfies (9.2.1) then there is a smooth function \( V : {\mathbb{R}}^{2n} \rightarrow \mathbb{R} \) such that \( \left( {{x}_{1},{y}_{1}}\right) = \psi \left( {{x}_{0},{y}_{0}}\right) \) if and only if\n\n\[ \n{x}_{1} - {x}_{0} = \frac{\partial V}{\partial y}\left( {{x}_{1},{y}_{0}}\right) ,\... | Proof: By assumption there exists a smooth map\n\n\[ \n{\mathbb{R}}^{2n} \rightarrow {\mathbb{R}}^{n} : \left( {{x}_{1},{y}_{0}}\right) \mapsto {x}_{0} = f\left( {{x}_{1},{y}_{0}}\right) \n\]\n\nsuch that \( {x}_{0} = f\left( {{x}_{1},{y}_{0}}\right) = f\left( {u\left( {{x}_{0},{y}_{0}}\right) ,{y}_{0}}\right) \) . for... | No |
A symplectic matrix \( \Psi \in \operatorname{Sp}\left( {2n}\right) \) of the form (9.1.2) admits a generating function \( V = V\left( {{x}_{1},{y}_{0}}\right) \) if and only if \( \det \left( A\right) \neq 0 \) . | If this holds then a generating function is given by\n\n\[ V\left( {{x}_{1},{y}_{0}}\right) = - \frac{1}{2}\left\langle {{x}_{1}, C{A}^{-1}{x}_{1}}\right\rangle + \left\langle {{y}_{0},\left( {\mathbb{1} - {A}^{-1}}\right) {x}_{1}}\right\rangle + \frac{1}{2}\left\langle {{y}_{0},{A}^{-1}B{y}_{0}}\right\rangle \]\n\nfor... | Yes |
Lemma 9.2.4 A point \( \mathbf{z} \in \mathcal{P} \) is critical for \( \Phi \) with respect to variations of the form \( \mathbf{z} + \zeta \in \mathcal{P} \) with \( {\xi }_{0} = {\xi }_{N} = 0 \) if and only if \( \mathbf{z} \) satisfies the Hamiltonian difference equations (9.2.3) (the first equation for \( 0 \leq ... | Proof: The derivatives of \( \Phi \) are given by\n\n\[ \frac{\partial \Phi }{\partial {x}_{j}} = {y}_{j - 1} - {y}_{j} - \frac{\partial {V}_{j - 1}}{\partial x}\left( {{x}_{j},{y}_{j - 1}}\right) \]\n\n\[ \frac{\partial \Phi }{\partial {y}_{j}} = {x}_{j + 1} - {x}_{j} - \frac{\partial {V}_{j}}{\partial y}\left( {{x}_{... | Yes |
Lemma 9.2.6 A periodic sequence \( \mathbf{z} \) is a critical point of \( \Phi : {\mathcal{P}}_{\text{per }} \rightarrow \mathbb{R} \) if and only if \( {z}_{0} = \left( {{x}_{0},{y}_{0}}\right) \) is a fixed point of \( \psi \) and \( {z}_{j + 1} = {\psi }_{j}\left( {z}_{j}\right) \) for all \( j \in \mathbb{Z} \) . ... | Proof: To prove the first assertion, note that the identities in the proof of Lemma 9.2.4 continue to hold for all \( j \in \mathbb{Z} \) . Hence \( \mathbf{z} \in {\mathcal{P}}_{\text{per }} \) is a critical point of \( \Phi \) if and only if \( {z}_{j + 1} = {\psi }_{j}\left( {z}_{j}\right) \) for all \( j \in \mathb... | Yes |
Proposition 9.3.1 Suppose \( \omega = - {d\lambda } \) and \( {\phi }_{t} \in \operatorname{Diff}\left( M\right) \) is an isotopy starting at the identity \( {\phi }_{0} = \mathrm{{id}} \) . Then \( {\phi }_{t} \) is a symplectic isotopy if and only if the 1 -form \( {\phi }_{t}^{ * }\lambda - \lambda \) is closed for ... | Proof: Let \( {\phi }_{t} \) be a Hamiltonian isotopy generated by Hamiltonian vector fields \( {X}_{t} : M \rightarrow {TM} \) as in equation (9.3.1). Then\n\n\[{\partial }_{t}{\phi }_{t}^{ * }\lambda = {\phi }_{t}^{ * }\left( {\iota \left( {X}_{t}\right) {d\lambda } + {d\iota }\left( {X}_{t}\right) \lambda }\right) =... | Yes |
Corollary 9.3.3 Let \( \left( {M,\omega }\right) \) be an exact symplectic manifold and let \( {\phi }_{t} \) be a symplectic isotopy. If \( {\phi }_{t} \) is Hamiltonian for each \( t \) then \( {\phi }_{t} \) is a Hamiltonian isotopy, i.e. it is generated by Hamiltonian vector fields. | Observe that the value of the function \( {F}_{t} \) in (9.3.2) at a point \( x \in M \) is the integral of the action form \( \lambda - {H}_{s}{ds} \) along the path \( \left\lbrack {0, t}\right\rbrack \rightarrow M : s \mapsto {\phi }_{s}\left( x\right) \) . This action integral made its first appearance in Section 1... | No |
Corollary 9.3.6 Let \( \left( {M, - {d\lambda }}\right) \) be an exact symplectic manifold, let \( \iota : L \rightarrow M \) be an exact Lagrangian embedding, and let \( \phi \in \operatorname{Ham}\left( M\right) \) . Then \( \phi \circ \iota : L \rightarrow M \) is an exact Lagrangian embedding. | Proof: \( {\left( \phi \circ \iota \right) }^{ * }\lambda = {\iota }^{ * }\left( {{\phi }^{ * }\lambda - \lambda }\right) + {\iota }^{ * }\lambda \) is exact by Proposition 9.3.1. | Yes |
Corollary 9.3.7 Let \( \\left( {M,\\omega }\\right) \) be an exact symplectic manifold, let \( \\Delta \\subset M \\times M \) denote the diagonal, and let \( \\alpha \\in {\\Omega }^{1}\\left( {M \\times M}\\right) \) be a 1 -form that satisfies\n\n\[ \n- {d\\alpha } = \\left( {-\\omega }\\right) \\oplus \\omega ,{\\l... | Proof: The inclusion of the diagonal \( \\iota : M \\rightarrow M \\times M \) is an exact Lagrangian embedding of \( M \) into \( \\left( {M \\times M, - {d\\alpha }}\\right) \) because \( {\\iota }^{ * }\\alpha = 0 \) and \( \\psi \\mathrel{\\text{:=}} \\mathrm{{id}} \\times \\phi \) is a Hamiltonian symplectomorphis... | Yes |
Proposition 9.3.10 Let \( \left( {M,\omega }\right) \) be an exact manifold with \( {H}^{1}\left( {M;\mathbb{R}}\right) = 0 \) and let \( \phi \in {\operatorname{Ham}}_{\mathrm{c}}\left( M\right) \) be a Hamiltonian symplectomorphism with compact support. Then all its generating functions \( {S}_{\alpha ,\phi } \) of c... | Proof: First consider the case where \( \alpha = \left( {-\lambda }\right) \oplus \lambda \) with \( \omega = - {d\lambda } \) and suppose that \( {\alpha }^{\prime } = \left( {-{\lambda }^{\prime }}\right) \oplus {\lambda }^{\prime } \) is another such form. Then \( \beta \mathrel{\text{:=}} {\lambda }^{\prime } - \la... | Yes |
This example shows that the condition \( {H}^{1}\left( {M;\mathbb{R}}\right) = 0 \) in Proposition 9.3.10 cannot be removed. Let \( M \mathrel{\text{:=}} \mathbb{R}/\mathbb{Z} \times \mathbb{R} \cong {T}^{ * }{S}^{1} \) be the cotangent bundle of the circle with coordinates \( x \in \mathbb{R}/\mathbb{Z} \) and \( y \i... | \[ \dot{x} = {h}^{\prime }\left( y\right) ,\;\dot{y} = 0 \] and the Hamiltonian symplectomorphism \( \phi \) is given by \( \phi \left( {x, y}\right) = \left( {x + {h}^{\prime }\left( y\right), y}\right) \) . Consider the 1-forms \( {\lambda }_{r} \mathrel{\text{:=}} \left( {y + r}\right) {dx} \) with \( - d{\lambda }_... | Yes |
Let \( \left( {M,\omega }\right) \) be an exact manifold with \( {H}^{1}\left( {M;\mathbb{R}}\right) = 0 \) and let \( \phi : M \rightarrow M \) be a compactly supported Hamiltonian symplectomorphism. Choose a 1 -form \( \lambda \in {\Omega }^{1}\left( M\right) \) such that \( \omega = - {d\lambda } \) and a Hamiltonia... | Proof: Proposition 9.3.1 and Proposition 9.3.10. | No |
Corollary 9.3.14 Let \( \left( {M,\omega }\right) \) be a connected, simply connected, and exact symplectic manifold and let \( {\phi }_{t} \in {\operatorname{Ham}}_{\mathrm{c}}\left( M\right) \) be the compactly supported Hamiltonian isotopy generated by \( {H}_{t} \in {C}_{0}^{\infty }\left( {M,\mathbb{R}}\right) \) ... | Proof: This is an obvious consequence of the fact that whenever \( z = {\phi }_{t}\left( z\right) \) for all \( t \) then \( {X}_{t}\left( {{\phi }_{t}\left( z\right) }\right) = 0 \) for all \( t \) and hence the formula (9.3.5) shows that\n\n\[ \n{\mathcal{A}}_{{\phi }_{1}}\left( z\right) = - {\int }_{0}^{1}{H}_{t}\le... | Yes |
Proposition 9.3.17 Let \( \left( {M,\omega }\right) \) be a symplectic manifold without boundary, let \( U \subset M \times M \) be an open neighbourhood of the diagonal, and let \( \alpha \in {\Omega }^{1}\left( U\right) \) satisfy (9.3.4) on \( U \) . Let \( {\phi }_{t} \in \operatorname{Ham}\left( {M,\omega }\right)... | Proof: Define \( {U}_{t} \mathrel{\text{:=}} {\left( \mathrm{{id}} \times {\phi }_{t}\right) }^{-1}\left( U\right) \cap U \) and consider the Hamiltonian isotopy \( {\psi }_{t} \mathrel{\text{:=}} \mathrm{{id}} \times {\phi }_{t} : {U}_{t} \rightarrow U \) . Proposition 9.3.1 extends to this setting and so there is a s... | Yes |
Lemma 9.4.1 Let \( {\psi }_{t} : {T}^{ * }L \rightarrow {T}^{ * }L \) be a Hamiltonian isotopy. Then the restriction of its time- \( 1 \) map \( \psi \) to the zero section induces an exact Lagrangian embedding \( \psi \circ \iota : L \rightarrow {T}^{ * }L \) . | Proof: Recall from Proposition 9.3.1 that every Hamiltonian symplectomor-phism \( \psi = {\psi }_{1} \) of a cotangent bundle \( {T}^{ * }L \) satisfies\n\n\[{\psi }_{t}^{ * }{\lambda }_{\text{can }} - {\lambda }_{\text{can }} = d{F}_{t},\;{F}_{t} \mathrel{\text{:=}} {\int }_{0}^{t}\left( {\iota \left( {X}_{s}\right) {... | Yes |
Proposition 9.4.2 Let \( {\psi }_{t} \) be a Hamiltonian isotopy of \( {T}^{ * }L \), generated by the Hamiltonian functions \( {H}_{t} : {T}^{ * }L \rightarrow \mathbb{R} \), such that \( \Lambda \mathrel{\text{:=}} {\psi }_{1}\left( {L}_{0}\right) \subset {T}^{ * }L \) is a section of the cotangent bundle. Define the... | Proof: Denote \( \psi = {\psi }_{1} \) and note first that the map\n\n\[ f = \pi \circ \psi \circ \iota : L \rightarrow L \]\n\nis a diffeomorphism (and hence \( \Lambda = \psi \left( {L}_{0}\right) \) is a section of \( {T}^{ * }L \) ) if and only if the boundary value problem (9.4.2) has a unique solution \( \gamma \... | Yes |
Proposition 9.4.4 If the variational family \( \left( {E,\Phi }\right) \) is transversal then the set \( \mathcal{C}\left( {E,\Phi }\right) \) of fibre critical points is a manifold of dimension \( n = \dim L \) and the map \( {\iota }_{\Phi } : \mathcal{C}\left( {E,\Phi }\right) \rightarrow {T}^{ * }L \) is an exact L... | Proof: The fibre normal bundle \( {N}_{E} \) is a coisotropic submanifold of \( {T}^{ * }E \) of codi-mension \( N \mathrel{\text{:=}} \dim E - \dim L \) . The isotropic leaves are the fibres of \( E \) and hence the symplectic quotient (see Proposition 5.4.5) of \( {N}_{E} \) can be identified with \( {T}^{ * }L \) . ... | Yes |
Lemma 9.4.5 Let \( A = {A}^{\mathrm{T}} \in {\mathbb{R}}^{n \times n} \) and \( C = {C}^{\mathrm{T}} \in {\mathbb{R}}^{N \times N} \) be symmetric matrices and \( B \in {\mathbb{R}}^{n \times N} \) . Then\n\n\[ \Lambda \mathrel{\text{:=}} \left\{ {\left( {x,{Ax} + {B\xi }}\right) \mid {B}^{\mathrm{T}}x + {C\xi } = 0}\r... | Proof: Note that \( \left( {x, y}\right) \in {\Lambda }^{\omega } \) if and only if\n\n\[ {B}^{\mathrm{T}}{x}^{\prime } + C{\xi }^{\prime } = 0\; \Rightarrow \;\left\langle {{x}^{\prime }, y}\right\rangle - \left\langle {A{x}^{\prime } + B{\xi }^{\prime }, x}\right\rangle = 0. \]\n\nThis can be rephrased as\n\n\[ {B}^{... | Yes |
Example 9.4.8 Consider the case where \( E \) is the set of all paths \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow {T}^{ * }L \) with \( \gamma \left( 0\right) \in {L}_{0} \), and the projection \( \pi : E \rightarrow L \) is given by\n\n\[ \pi \left( \gamma \right) \mathrel{\text{:=}} {\pi }_{{T}^{ * }L}\le... | Moreover, at a fibre critical point the formula (9.4.3) holds with \( {v}^{ * } \) equal to the vertical component of \( \gamma \left( 1\right) \) . Hence the Lagrangian submanifold of \( {T}^{ * }L \) generated by \( \Phi \) is the image of the zero section \( {L}_{0} \subset {T}^{ * }L \) under the time-1 map of the ... | Yes |
Proposition 10.1.2 The Hamiltonian symplectomorphisms form a path-connected normal subgroup \( \operatorname{Ham}\left( M\right) \) of \( \operatorname{Symp}\left( M\right) \) . | Proof: See Exercise 3.1.14. That \( \operatorname{Ham}\left( M\right) \) is path-connected follows from its very definition. | No |
Lemma 10.2.1 The right-hand side of (10.2.2) depends only on the homotopy class of \( \gamma \) and on the homotopy class of \( {\psi }_{t} \) with fixed endpoints. | Proof: Since \( {\psi }_{t} \) is a family of symplectomorphisms, the 1-forms \( \iota \left( {X}_{t}\right) \omega \) are closed. Hence the right-hand side of (10.2.2) depends only on the homotopy class of \( \gamma \) . Now define the map \( \beta : \mathbb{R}/\mathbb{Z} \times \left\lbrack {0,1}\right\rbrack \righta... | Yes |
Lemma 10.2.2\n\[ \int {\beta }^{ * }\omega = \int {\beta }^{\prime * }\omega \] | Proof: It suffices to shows that the maps\n\n\[ {\psi }_{1}\left( {\beta \left( {s, t}\right) }\right) = {\psi }_{1} \circ {\psi }_{t}^{-1}\left( {\gamma \left( s\right) }\right) ,\;{\beta }^{\prime }\left( {1 - t, s}\right) = {\psi }_{1 - t}\left( {\gamma \left( s\right) }\right) \]\n\nare homotopic with fixed boundar... | Yes |
Lemma 10.2.7 If \( \omega = - {d\lambda } \) and \( {\psi }_{t} : M \rightarrow M \) is a compactly supported symplectic isotopy then \( \operatorname{Flux}\left( \left\{ {\psi }_{t}\right\} \right) = \left\lbrack {\lambda - {\psi }_{1}^{ * }\lambda }\right\rbrack \) . | Proof: If \( {\psi }_{t} \) is a symplectic isotopy generated by compactly supported symplectic vector fields \( {X}_{t} \), then\n\n\[ \left\lbrack {\iota \left( {X}_{t}\right) \omega }\right\rbrack = - \left\lbrack {{\psi }_{t}^{ * }\iota \left( {X}_{t}\right) {d\lambda }}\right\rbrack = - \left\lbrack {{\psi }_{t}^{... | Yes |
Lemma 10.2.11 If \( \psi \in {\operatorname{Symp}}_{0}\left( {M,\omega }\right) \) is sufficiently close to the identity in the \( {C}^{1} \) -topology and \( \sigma = \mathcal{C}\left( \psi \right) \in {\Omega }^{1}\left( M\right) \) is defined as above, then\n\n\[ \psi \in \operatorname{Ham}\left( {M,\omega }\right) ... | Proof: Consider the symplectic isotopy \( {\psi }_{t} \) with \( \mathcal{C}\left( {\psi }_{t}\right) = {t\sigma } \) . Then it follows from Lemma 10.2.8 that \( \operatorname{Flux}\left( \left\{ {\psi }_{t}\right\} \right) = - \left\lbrack \sigma \right\rbrack \) . Since \( \psi \) is a Hamiltonian symplectomor-phism,... | Yes |
Proposition 10.2.12 Every smooth path \( {\psi }_{t} \in \operatorname{Ham}\left( M\right) \) is generated by Hamiltonian vector fields. | Proof: Assume \( {\psi }_{0} = \) id. Then, by Lemma 10.2.11, the cohomology class of \( {\sigma }_{t} = \mathcal{C}\left( {\psi }_{t}\right) \) is in \( {\Gamma }_{\omega } = \operatorname{Flux}\left( {{\pi }_{1}\left( {{\operatorname{Symp}}_{0}\left( M\right) }\right) }\right) \) for \( t \) sufficiently small. Since... | Yes |
Proposition 10.2.13 Let \( \left( {M,\omega }\right) \) be a closed, connected symplectic manifold.\n\n(i) There is an exact sequence of simply connected Lie groups\n\n\[ 0 \rightarrow \widetilde{\operatorname{Ham}}\left( {M,\omega }\right) \rightarrow {\widetilde{\operatorname{Symp}}}_{0}\left( {M,\omega }\right) \rig... | Proof: By Proposition 10.2.12, every smooth path \( {\psi }_{t} \in \operatorname{Ham}\left( M\right) \) which starts at the identity is a Hamiltonian isotopy and hence has zero flux. This shows that \( \widetilde{\operatorname{Ham}}\left( M\right) \subset \ker \left( \text{Flux}\right) \) . Conversely, Theorem 10.2.5 ... | Yes |
Theorem 10.2.17. (Ono) Let \( \left( {M,\omega }\right) \) be a closed symplectic manifold. Then the group \( \operatorname{Ham}\left( {M,\omega }\right) \) is \( {C}^{1} \) -closed in \( {\operatorname{Symp}}_{0}\left( {M,\omega }\right) \) . | Proof: See Ono [516]. | No |
Proposition 10.2.19 If \( \left( {M,\omega }\right) \) is a closed symplectic manifold and \( {\left\{ {\phi }_{t}\right\} }_{t \in \mathbb{R}/\mathbb{Z}} \) is a loop in \( \operatorname{Ham}\left( {M,\omega }\right) \), then the loops \( t \mapsto {\phi }_{t}\left( x\right) \) are null-homologous. | Proof: We examine the Poincaré dual of the homology class of the loop\n\n\[ \gamma : \mathbb{R}/\mathbb{Z} \rightarrow M,\;\gamma \left( t\right) \mathrel{\text{:=}} {\phi }_{t}\left( {p}_{0}\right) . \]\n\nLet \( Q \) be a closed oriented \( \left( {{2n} - 1}\right) \) -manifold and \( f : Q \rightarrow M \) be a smoo... | Yes |
Proposition 10.2.21 Let \( {\phi }_{t} \in {\operatorname{Symp}}_{0}\left( {{\mathbb{T}}^{2n},{\omega }_{0}}\right) \) be a symplectic isotopy with a lift \( {\widetilde{\phi }}_{t} : {\mathbb{R}}^{2n} \rightarrow {\mathbb{R}}^{2n} \) such that\n\n\[ \n{\widetilde{\phi }}_{t}\left( {w + \ell }\right) = {\widetilde{\phi... | Proof: Let \( {H}_{t} \) be a smooth family of generating Hamiltonians for \( {\phi }_{t} \), so that \n\n\[ \n\frac{d}{dt}{\widetilde{\phi }}_{t} = - {J}_{0}\nabla {H}_{t} \circ {\widetilde{\phi }}_{t} \n\] \n\nWe claim that there are functions \( {h}_{j}\left( t\right) \) such that \n\n\[ \n{H}_{t}\left( {w + \ell }\... | Yes |
Theorem 10.3.1 (Banyaga) If \( \left( {M,\omega }\right) \) is a closed connected symplectic manifold then the group \( \operatorname{Ham}\left( {M,\omega }\right) \) is simple, i.e. it contains no nontrivial normal subgroups. | ## Proof: See Banyaga \( \left\lbrack {{50},{51}}\right\rbrack \) . | No |
Lemma 10.3.3 The number \( \operatorname{CAL}\left( \phi \right) \) is independent of the 1 -form \( \lambda \in {\Omega }^{1}\left( M\right) \) with \( \omega = - {d\lambda } \) used to define it. Moreover, the map \[ \text{CAL :}{\operatorname{Ham}}_{\mathrm{c}}\left( M\right) \rightarrow \mathbb{R} \] is a homomorph... | Proof: Let \( \lambda + \alpha \) be any other such 1 -form with \( {d\alpha } = 0 \) . Then, since \( \phi \) induces the identity map on compactly supported cohomology, there exists a compactly supported smooth function \( G : M \rightarrow \mathbb{R} \) such that \( {\phi }^{ * }\alpha - \alpha = {dG} \) . We must p... | Yes |
Lemma 10.3.4 Let \( \phi \in {\operatorname{Ham}}_{\mathrm{c}}\left( {M, - {d\lambda }}\right) \). Then, using the above notation, we have\n\[ \operatorname{CAL}\left( \phi \right) = {\int }_{0}^{1}{\int }_{M}{H}_{t}{\omega }^{n}{dt} \] and \[ \operatorname{CAL}\left( \phi \right) = - \frac{1}{n + 1}{\int }_{M}{\phi }^... | Proof: For each \( t \) let \( {F}_{t} : M \rightarrow \mathbb{R} \) be the unique compactly supported function such that \[ {\phi }_{t}^{ * }\lambda - \lambda = d{F}_{t} \] Then, by a simple calculation \[ {\int }_{M}{\phi }_{t}^{ * }\lambda \land \lambda \land {\omega }^{n - 1} = {\int }_{M}\left( {{\phi }_{t}^{ * }\... | Yes |
Theorem 10.3.7 (Banyaga) Let \( \left( {M,\omega }\right) \) be a noncompact connected symplectic manifold without boundary. Then the kernel of the Calabi homomorphism CAL : \( {\operatorname{Ham}}_{\mathrm{c}}\left( M\right) \rightarrow \mathbb{R}/\Lambda \) is a simple group. | Proof: See [50]. | No |
Theorem A. Let \( M \) be a closed symplectic four-manifold with Betti numbers\n\n\[ \n{b}_{1} \mathrel{\text{:=}} \dim {H}^{1}\left( {M;\mathbb{R}}\right) = 0,\;{b}_{2} \mathrel{\text{:=}} \dim {H}^{2}\left( {M;\mathbb{R}}\right) \geq 3.\n\]\n\nAssume \( M \) is minimal (i.e. it does not contain a symplectically embed... | (ii) The symplectomorphism \( \phi = {\tau }_{\Lambda }^{2} \) in Theorem A is smoothly isotopic to the identity by an isotopy localized near \( \Lambda \) . Seidel computed the Floer cohomology group \( {\mathrm{{HF}}}^{ * }\left( {\tau }_{\Lambda }\right) \) with its module structure over the quantum cohomology ring,... | Yes |
Proposition 11.1.5 Let \( \psi : {\mathbb{T}}^{2n} \rightarrow {\mathbb{T}}^{2n} \) be an exact symplectomorphism which is sufficiently close to the identity in the \( {C}^{1} \) -topology. Then \( \psi \) has at least \( {2n} + 1 \) fixed points. If the fixed points of \( \psi \) are all nondegenerate then \( \psi \) ... | Proof: Let \( V : {\mathbb{T}}^{2n} \rightarrow \mathbb{R} \) be the generating function of Lemma 11.1.1. Then the critical points of \( V \) are the fixed points of \( \psi \) . But any function on the torus must have at least \( {2n} + 1 \) critical points. (If a function on a manifold has \( \ell \) critical points ... | Yes |
Proposition 11.1.8 The Arnold conjecture holds for every Hamiltonian symplectomorphism \( \psi \) of a closed symplectic manifold \( \left( {M,\omega }\right) \) which is sufficiently close to the identity in the \( {C}^{1} \) -topology. | Proof: Recall from Proposition 3.4.14 that the graph of \( \psi \) is a Lagrangian submanifold of \( M \times M \) which is close to the diagonal \( \Delta \) . Also a neighbourhood of \( \Delta \) may be identified with a neighbourhood of the zero section in \( {T}^{ * }M \) via a symplectomorphism \( \Psi \) . With t... | Yes |
Theorem 11.1.9 (Conley-Zehnder) Every Hamiltonian symplectomorphism of the standard torus \( {\mathbb{T}}^{2n} \) has at least \( {2}^{2n} \) geometrically distinct fixed points provided that these are all nondegenerate. In the degenerate case the number of fixed points is at least \( {2n} + 1 \) . | Proof: See pages 430 and 434. | No |
Lemma 11.2.2 If \( \left( {{N}_{\alpha },{L}_{\alpha }}\right) \) and \( \left( {{N}_{\beta },{L}_{\beta }}\right) \) are index pairs for \( \Lambda \) then the index spaces \( {N}_{\alpha }/{L}_{\alpha } \) and \( {N}_{\beta }/{L}_{\beta } \) are homotopy equivalent. | Proof: The proof is an elementary homotopy argument. For each \( t \geq 0 \), the flow determines a (possibly discontinuous) map \( {\phi }_{\beta \alpha }^{t} : {N}_{\alpha }/{L}_{\alpha } \rightarrow {N}_{\beta }/{L}_{\beta } \) defined by\n\n\[ \n{\phi }_{\beta \alpha }^{t}\left( x\right) \mathrel{\text{:=}} \left\{... | Yes |
As an example consider a hyperbolic fixed point \( x = 0 \) of a differential equation\n\n\[ \dot{x} = v\left( x\right) \]\n\nin \( {\mathbb{R}}^{n} \) . Denote by \( {E}^{s} \) and \( {E}^{u} \) the stable and unstable subspaces of the linearized system \( \dot{\xi } = {dv}\left( 0\right) \xi \) . Thus \( {E}^{s} \) i... | It follows that \( N/L \) has the homotopy type of a pointed \( k \) -sphere, where \( k = \dim {E}^{u} \) is the index of the hyperbolic fixed point and the index polynomial is \( {p}_{\Lambda }\left( s\right) = {s}^{k} \) . | Yes |
Theorem 11.2.4 (Morse inequalities) For \( k = 0,\ldots, n \) , \[ {c}_{k}\left( \Lambda \right) - {c}_{k - 1}\left( \Lambda \right) + \cdots \pm {c}_{0}\left( \Lambda \right) \geq {b}_{k}\left( \Lambda \right) - {b}_{k - 1}\left( \Lambda \right) + \cdots \pm {b}_{0}\left( \Lambda \right) \] and equality holds for \( k... | Proof: Fix an index pair \( \left( {N, L}\right) \) for \( \Lambda \) . For every regular value \( a \in \mathbb{R} \) of \( {\left. \Phi \right| }_{N} \) define \[ {N}^{a} = \{ x \in N \mid \Phi \left( x\right) \leq a\} \cup L. \] Then for every critical value \( c \) of \( {\left. \Phi \right| }_{N} \) the set of cri... | Yes |
The Ljusternik-Schnirelmann category of an \( n \) -dimensional smooth manifold \( M \) is less than or equal to \( n + 1 \). | To see this, triangulate \( M \) and then consider the first barycentric subdivision \( {\mathcal{T}}_{1} \) of this triangulation \( \mathcal{T} \) . This is the triangulation whose vertices consist of all the vertices in \( \mathcal{T} \) plus the barycentres of all the faces in \( \mathcal{T} \) . Let \( {\mathcal{V... | Yes |
Lemma 11.2.8 For every compact metric space \( M \)\n\n\[ {\nu }_{\mathrm{{LS}}}\left( M\right) \geq {cl}\left( M\right) \] | Proof: We prove the lemma in the case where \( M \) is a manifold and \( {H}^{ * } \) denotes the de Rham cohomology. The argument can be easily adapted to the general case. Assume that\n\n\[ M \subset {U}_{1} \cup \cdots \cup {U}_{N} \]\n\nwhere the sets \( {U}_{j} \) are all cohomologically trivial. We must prove tha... | Yes |
Theorem 11.2.9 (Ljusternik-Schnirelmann) Let \( M \) be a compact metric space and denote by \( {cl}\left( M\right) \) the cuplength of \( M \) with respect to any cohomology theory. Then a gradient-like flow on \( M \) has at least \( {cl}\left( M\right) \) constant solutions. | Proof: Let \( \nu \) be any category on \( M \) . We prove that a gradient-like flow on \( M \) has at least \( \nu \left( M\right) \) constant orbits. Then the theorem follows from Lemma 11.2.8.\n\nGiven any number \( c > 0 \) define\n\n\[ \n{M}^{c} = \{ x \in M \mid \Phi \left( x\right) \leq c\} .\n\]\n\nIf \( c \) i... | Yes |
Theorem 11.3.8 Let \( L \) be a closed manifold and let \( \psi : {T}^{ * }L \rightarrow {T}^{ * }L \) be a Hamiltonian symplectomorphism. Then \( \# \left( {\psi \left( L\right) \cap L}\right) \geq {cl}\left( L\right) \) . If \( \psi \left( L\right) \) intersects \( L \) transversally then \( \# \left( {\psi \left( L\... | Proof: See page 439. | No |
Theorem 11.3.10 (Gromov) Let \( L \) be a closed \( n \) -manifold and \( \Lambda \subset {T}^{ * }L \) be a compact exact Lagrangian submanifold without boundary. Then\n\n\[ \Lambda \cap L \neq \varnothing \]\n\nand\n\n\[ \Lambda \cap \psi \left( \Lambda \right) \neq \varnothing \]\n\nfor every Hamiltonian symplectomo... | Proof: See Gromov [287], Audin-Lalonde-Polterovich [35], and McDuff-Sala-mon [470, Theorem 9.2.16 and Corollary 9.2.17]. All known proofs of this result use \( J \) -holomorphic curves or Floer homology and go beyond the scope of this book. | No |
Theorem 12.1.1 (Nonsqueezing theorem) If there exists a symplectic embedding of \( \\left( {{B}^{2n}\\left( r\\right) ,{\\omega }_{0}}\\right) \) into \( \\left( {{Z}^{2n}\\left( R\\right) ,{\\omega }_{0}}\\right) \) then \( r \\leq R \) . | Proof: See page 484. | No |
For \( a > 0 \) denote by \( {S}^{2}\left( a\right) \) the 2-sphere of area \( a \) and by \( {\mathbb{T}}^{2}\left( a\right) \) the 2-torus of area \( a \) . It follows from Gromov’s proof of the Nonsqueezing Theorem and from Remark 12.1.3 that for \( r \leq R \)\n\n\[ \n{w}_{G}\left( {{S}^{2}\left( {\pi {r}^{2}}\righ... | Moreover, when \( n = 1 \) it follows from part (ii) of Remark 12.1.4 that we may even choose \( R = r/\sqrt{2} \) without decreasing the capacity:\n\n\[ \n{w}_{G}\left( {{S}^{2}\left( {\pi {r}^{2}}\right) \times {\mathbb{T}}^{2}\left( {\pi {r}^{2}/2}\right) }\right) = \pi {r}^{2},\n\] | Yes |
For every symplectic manifold \( \left( {M,\omega }\right) \) the group of symplec-tomorphisms of \( \left( {M,\omega }\right) \) is \( {C}^{0} \) -closed in the group of all diffeomorphisms of \( M \) . | See page 464 for a proof based on Theorem 12.1.1. | No |
Proposition 12.2.2 Let \( \psi : {\mathbb{R}}^{2n} \rightarrow {\mathbb{R}}^{2n} \) be a diffeomorphism and let \( c \) be a symplectic capacity on \( {\mathbb{R}}^{2n} \) which satisfies (12.1.1). Then the following are equivalent. (i) \( \psi \) preserves the capacity of ellipsoids, i.e. it satisfies \( c\left( {\psi... | Proof: That (ii) implies (i) is obvious. The converse is proved on page 464. | No |
Lemma 12.2.3 Let \( c \) be a symplectic capacity on \( {\mathbb{R}}^{2n} \) which satisfies (12.1.1). Let \( {\psi }_{\nu } : {\mathbb{R}}^{2n} \rightarrow {\mathbb{R}}^{2n} \) be a sequence of continuous maps converging to a homeomorphism \( \psi : {\mathbb{R}}^{2n} \rightarrow {\mathbb{R}}^{2n} \), uniformly on comp... | Proof: Without loss of generality we consider only ellipsoids centred at zero. We first prove that for every ellipsoid \( E \) and every positive number \( \lambda < 1 \) there exists a \( {\nu }_{0} > 0 \) such that\n\n\[{\psi }_{\nu }\left( {\lambda E}\right) \subset \psi \left( E\right) \subset {\psi }_{\nu }\left( ... | Yes |
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