Let $\omega \in \mathrm { A } ( E )$ and $\mathcal { B } = \left( b _ { 1 } , \ldots , b _ { n } \right)$ a basis of $E$. (a) Show that there exists a unique matrix $M \in \mathcal { M } _ { n } ( \mathbb { R } )$, whose coefficients we shall specify, such that for all $( x , y ) \in E ^ { 2 } , \omega ( x , y ) = { } ^ { t } X M Y$ where $X , Y \in \mathbb { R } ^ { n }$ are the column matrices representing respectively $x$ and $y$ in the basis $\mathcal { B }$: $$X = \left( \begin{array} { c } x _ { 1 } \\ \vdots \\ x _ { n } \end{array} \right) , \quad Y = \left( \begin{array} { c } y _ { 1 } \\ \vdots \\ y _ { n } \end{array} \right) , \quad \begin{aligned} & x = x _ { 1 } b _ { 1 } + \cdots + x _ { n } b _ { n } \\ & y = y _ { 1 } b _ { 1 } + \cdots + y _ { n } b _ { n } . \end{aligned}$$ We then denote $M = \operatorname { Mat } _ { \mathcal { B } } ( \omega )$. (b) Show that $M$ is antisymmetric, that is, ${ } ^ { t } M = - M$. (c) Show that the vector space $\mathrm { A } ( E )$ is of dimension 1 when $E$ is of dimension 2. (d) Show the equivalence between the three following statements. $\left( \mathcal { E } _ { 1 } \right) : \quad \omega$ is a symplectic form on $E$. $\left( \mathcal { E } _ { 2 } \right) : \quad$ For all $x \in E \backslash \{ 0 \}$, there exists $y \in E$ such that $\omega ( x , y ) \neq 0$. $\left( \mathcal { E } _ { 3 } \right) : \quad \operatorname { Mat } _ { \mathcal { B } } ( \omega )$ is invertible.
Show that the map $\omega _ { 0 }$ defined by $$\begin{array} { r l c c } \omega _ { 0 } : & \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } & \rightarrow & \mathbb { R } \\ ( X , Y ) & \mapsto & { } ^ { t } X J _ { n } Y \end{array}$$ is a symplectic form on $\mathbb { R } ^ { n }$.
We fix a symplectic form $\omega$ on $E$. The purpose of questions 6 to 9 is to show that there exists a basis $\mathcal { B }$ of $E$ such that $\operatorname { Mat_{\mathcal {B}} } ( \omega ) = J _ { n }$. Treat the case where $E$ is of dimension 2.
We fix a symplectic form $\omega$ on $E$. Let $F$ be a vector subspace of $E$. (a) Show that, for every linear form $u : F \rightarrow \mathbb { R }$, there exists a linear form $\widetilde { u } : E \rightarrow \mathbb { R }$ whose restriction to $F$ coincides with $u$. We denote by $F ^ { \omega }$ the vector subspace of $E$ defined by $$F ^ { \omega } = \{ x \in E : \forall y \in F , \omega ( x , y ) = 0 \}$$ and $\psi _ { F }$ the linear map defined by $$\left\lvert \, \begin{aligned} \psi _ { F } : \quad E & \rightarrow F ^ { * } \\ x & \left. \mapsto \varphi _ { \omega } ( x ) \right| _ { F } \end{aligned} \right.$$ where $\left. \varphi _ { \omega } ( x ) \right| _ { F }$ is the restriction of $\varphi _ { \omega } ( x )$ to $F$. (b) Show that the restriction of $\omega$ to $F \times F$ is a symplectic form on $F$ if and only if $F \cap F ^ { \omega } = \{ 0 \}$. (c) What are the kernel and image of $\psi _ { F }$ ? (d) Show that $\operatorname { dim } ( F ) + \operatorname { dim } \left( F ^ { \omega } \right) = \operatorname { dim } ( E )$. (e) Show that, if the restriction of $\omega$ to $F \times F$ is a symplectic form on $F$, then $E = F \oplus F ^ { \omega }$ and the restriction of $\omega$ to $F ^ { \omega } \times F ^ { \omega }$ is a symplectic form on $F ^ { \omega }$.
We fix a symplectic form $\omega$ on $E$. Conclude that there exists a basis $\mathcal { B }$ of $E$ such that $\operatorname { Mat_{\mathcal {B}} } ( \omega ) = J _ { n }$. Deduce that $\omega$ tames at least one complex structure on $E$.
Let $P , Q \in \mathbb { R } [ X ]$ be nonzero polynomials of respective degrees $p$ and $q$ strictly positive. Show that the linear map $L _ { P , Q }$ defined by $$\left\lvert \, \begin{array} { c c c } L _ { P , Q } : \quad \mathbb { R } _ { q - 1 } [ X ] \times \mathbb { R } _ { p - 1 } [ X ] & \rightarrow \quad \mathbb { R } _ { p + q - 1 } [ X ] \\ ( V , W ) & \mapsto V P + W Q \end{array} \right.$$ is an isomorphism if and only if $P$ and $Q$ are coprime in $\mathbb { R } [ X ]$.
Let $d \in \mathbb { N } ^ { * }$. Construct a map $$\left\lvert \, \begin{aligned} r : \quad \mathbb { R } _ { d } [ X ] & \rightarrow \mathbb { R } \\ P & \mapsto r ( P ) \end{aligned} \right.$$ polynomial in the coefficients of $P$, such that, if $r ( P )$ is nonzero, then the roots of $P$ in $\mathbb { C }$ are simple. Hint: You may use the previous question.
Let $d \in \mathbb { N } ^ { * }$ and $f$ a polynomial function on $\mathbb { R } ^ { d }$. Suppose that the function $f$ is nonzero. Show that $f ^ { - 1 } ( \mathbb { R } \backslash \{ 0 \} )$ is dense in $\mathbb { R } ^ { d }$. Hint: You may use the fact that a nonzero polynomial in one variable has only finitely many roots.
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$. Show that there exists a unique $u \in \mathrm { GL } ( E )$ such that $\omega _ { 1 } ( x , y ) = \omega ( u ( x ) , y )$ for all $( x , y ) \in E ^ { 2 }$. Show then that $u$ belongs to the set $\mathcal { S }$ defined by $$\mathcal { S } = \left\{ u \in \mathrm { GL } ( E ) : \forall ( x , y ) \in E ^ { 2 } , \omega ( x , u ( y ) ) = \omega ( u ( x ) , y ) \right\}$$
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We assume that $E$ is of dimension 4. Let $\mathcal { B }$ be a basis of $E$ such that $\operatorname { Mat } _ { \mathcal { B } } ( \omega ) = J _ { 4 }$. Let $U \in \mathcal { M } _ { 4 } ( \mathbb { R } )$ be the matrix of $u$ in the basis $\mathcal { B }$. (a) What relation is there between the matrices $J _ { 4 }$ and $U$ ? (b) Show that there exist $N \in \mathcal { M } _ { 2 } ( \mathbb { R } )$ and $\alpha , \beta \in \mathbb { R }$ such that $$U = \left( \begin{array} { c c } N & \alpha J _ { 2 } \\ \beta J _ { 2 } & { } ^ { t } N \end{array} \right)$$ (c) Determine, as a function of $N , \alpha$ and $\beta$, the coefficients of the polynomial $T$ defined by $T ( X ) = \operatorname { det } \left( N - X I _ { 2 } \right) + \alpha \beta$. Show that $T$ is an annihilating polynomial of $U$.
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $U \in \mathcal{M}_4(\mathbb{R})$ be the matrix of $u$ in $\mathcal{B}$. Show that $U$ is diagonalizable over $\mathbb { C }$. Deduce that there exist $\lambda \in \mathbb { C } \backslash \mathbb { R }$ and vectors $Z$ and $Y$ of $\mathbb { C } ^ { 4 }$ linearly independent over $\mathbb { C }$ such that $U Z = \lambda Z$ and $U Y = \overline{\lambda} Y$.
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $U \in \mathcal{M}_4(\mathbb{R})$ be the matrix of $u$ in $\mathcal{B}$. Let $\lambda \in \mathbb{C} \backslash \mathbb{R}$ and $Z, Y \in \mathbb{C}^4$ be as in question 15. Let $Z _ { 1 } , Z _ { 2 } , Y _ { 1 } , Y _ { 2 }$ be vectors of $\mathbb { R } ^ { 4 }$ such that $Z = Z _ { 1 } + i Z _ { 2 }$ and $Y = Y _ { 1 } + i Y _ { 2 }$. Let $\left( z _ { 1 } , z _ { 2 } , y _ { 1 } , y _ { 2 } \right) \in E ^ { 4 }$ have coordinates respectively $Z _ { 1 } , Z _ { 2 } , Y _ { 1 } , Y _ { 2 }$ in the basis $\mathcal { B }$. Show that $\widetilde { \mathcal { B } } : = \left( z _ { 1 } , z _ { 2 } , y _ { 1 } , - y _ { 2 } \right)$ is a basis of $E$.
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $z_1, z_2, y_1, y_2 \in E$ be as defined in question 16. Show that $$\begin{aligned} & \omega \left( z _ { 1 } , z _ { 2 } \right) = \omega \left( y _ { 1 } , y _ { 2 } \right) = 0 \\ & \omega \left( z _ { 1 } , y _ { 1 } \right) = - \omega \left( z _ { 2 } , y _ { 2 } \right) \\ & \omega \left( z _ { 1 } , y _ { 2 } \right) = \omega \left( z _ { 2 } , y _ { 1 } \right) \end{aligned}$$
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $Z, Y \in \mathbb{C}^4$ be eigenvectors as in question 15, with $z_1, z_2, y_1, y_2$ as defined in question 16. Show that, by replacing $Y$ with $\xi Y$ where $\xi \in \mathbb { C } \backslash \{ 0 \}$ is suitably chosen, we have $\omega \left( z _ { 1 } , y _ { 1 } \right) = - 1$ and $\omega \left( z _ { 1 } , y _ { 2 } \right) = 0$.
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\widetilde{\mathcal{B}} = (z_1, z_2, y_1, -y_2)$ be the basis constructed in questions 16--18, where $\operatorname{Mat}_{\widetilde{\mathcal{B}}}(\omega) = J_4$. Show that there exist $r > 0$ and $\theta \in \mathbb { R } \backslash \pi \mathbb { Z }$ such that $$\operatorname { Mat } _ { \widetilde { \mathcal { B } } } ( u ) = r \left( \begin{array} { c c } R _ { \theta } & 0 \\ 0 & R _ { - \theta } \end{array} \right)$$ where $R _ { \theta } = \left( \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right)$, and conclude that $$\operatorname { Mat } _ { \widetilde { \mathcal { B } } } ( \omega ) = J _ { 4 } \quad \text{and} \quad \operatorname { Mat } _ { \widetilde { \mathcal { B } } } \left( \omega _ { 1 } \right) = r \left( \begin{array} { c c } 0 & - R _ { - \theta } \\ R _ { \theta } & 0 \end{array} \right).$$
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We consider a polynomial $P \in \mathbb { R } [ X ]$ annihilating $u$ and a decomposition $P = P _ { 1 } \cdots P _ { r }$, where $r \in \mathbb { N } ^ { * }$ and $P _ { 1 } , \ldots , P _ { r }$ are polynomials pairwise coprime in $\mathbb { R } [ X ]$. We denote $F _ { j } = \operatorname { ker } \left[ P _ { j } ( u ) \right]$ for $j = 1 , \ldots , r$. Show that $E = F _ { 1 } \oplus \cdots \oplus F _ { r }$ and that $F _ { j }$ is stable under $u$ for $j = 1 , \ldots , r$.
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We consider a polynomial $P \in \mathbb { R } [ X ]$ annihilating $u$ and a decomposition $P = P _ { 1 } \cdots P _ { r }$, where $r \in \mathbb { N } ^ { * }$ and $P _ { 1 } , \ldots , P _ { r }$ are polynomials pairwise coprime in $\mathbb { R } [ X ]$. We denote $F _ { j } = \operatorname { ker } \left[ P _ { j } ( u ) \right]$ for $j = 1 , \ldots , r$. The notation $F^{\omega}$ is defined in question 7. Show that, for all $j$ and $k$ belonging to $\{ 1 , \ldots , r \}$ and distinct, we have $F _ { k } \subset F _ { j } ^ { \omega }$ and $F _ { k } \subset F _ { j } ^ { \omega _ { 1 } }$.
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We consider a polynomial $P \in \mathbb { R } [ X ]$ annihilating $u$ and a decomposition $P = P _ { 1 } \cdots P _ { r }$, where $r \in \mathbb { N } ^ { * }$ and $P _ { 1 } , \ldots , P _ { r }$ are polynomials pairwise coprime in $\mathbb { R } [ X ]$. We denote $F _ { j } = \operatorname { ker } \left[ P _ { j } ( u ) \right]$ for $j = 1 , \ldots , r$, and $F_1, \ldots, F_r$ are pairwise orthogonal for $\omega$ and for $\omega_1$ (as shown in question 21). Deduce that, for all $j \in \{ 1 , \ldots , r \}$, the restrictions of $\omega$ and $\omega _ { 1 }$ to $F _ { j } \times F _ { j }$ are symplectic forms on $F _ { j }$.
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. Suppose that the characteristic polynomial of $u$ has roots of multiplicity at most 2 in $\mathbb { C }$. Show that $E$ is the direct sum of subspaces of dimension 2 or 4, pairwise orthogonal for $\omega$ and $\omega _ { 1 }$, and on which the restrictions of $\omega$ and $\omega _ { 1 }$ are symplectic forms.
We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$. We consider the propositions: $\left( \mathcal { F } _ { 1 } \right) :$ There exists a complex structure tamed by $\omega$ and by $\omega _ { 1 }$. $\left( \mathcal { F } _ { 2 } \right) :$ The segment $\left[ \omega , \omega _ { 1 } \right] = \left\{ ( 1 - \theta ) \omega + \theta \omega _ { 1 } ; \theta \in [ 0,1 ] \right\}$ is included in the set of symplectic forms on $E$. Let $u$ be the automorphism of $E$ defined in question 13. Suppose that $\left( \mathcal { F } _ { 2 } \right)$ is satisfied and that the characteristic polynomial of $u$ has roots of multiplicity at most 2 in $\mathbb { C }$. Show that $( \mathcal { F } _ { 1 } )$ is satisfied. Hint: You may prove and then use the fact that, for all $\theta \in \mathbb { R } \backslash \pi \mathbb { Z }$, there exists $\phi \in \mathbb { R }$ such that, for all $X \in \mathbb { R } ^ { 2 } \backslash \{ 0 \} , { } ^ { t } X R _ { \phi } X > 0$ and ${ } ^ { t } X R _ { \theta + \phi } X > 0$.