Differential Forms on a Symplectic Manifold Let $M$ be a symplectic (algebraic) variety over a field $k$ of dimension $2n$ with a symplectic form $\omega$. Is the map $\Omega^{k}_{M} \to \Omega^{2n-k}_{M}$ given by a multiplication by $\omega^{n-k}$ an isomorphism? I think so but is there any reference?
 A: Just a hint (not sure about a reference, but it is a standard thing):
It is really a thing from linear algebra. If $V$ is a $2n$-dimensional vector space, and $\omega\in\bigwedge^2 V$ is non-degenerate, then the map $\bigwedge^k V\to\bigwedge^{2n-k}V$, $\alpha\mapsto\alpha\wedge\omega^{n-k}$ is bijective.
One way how to prove it is to introduce 3 linear operators $E,F,H$ on $\bigwedge V$. Namely, $E\alpha:=\omega\wedge\alpha$, $F\alpha:=i_\pi\alpha$ (here $\pi\in\bigwedge^2 V^*$ is the inverse of $\omega$), and finally, $H\alpha:=(m-n)\,\alpha$ for $\alpha\in\bigwedge^m V$. These then satisfy $[H,E]=2E$, $[H,F]=-2F$, $[E,F]=H$ (and it's possible that I'm missing some numerical coefficients in the definitions of $E,F,H$, but it's not essential).
In other words, we have an action of the Lie algebra $sl_2$ on $\bigwedge V$. And $\bigwedge^k V$ is the subspace of the vectors of weight $-(n-k)$ and $\bigwedge^{2n-k}V$ the subspace of the vectors of weight $+(n-k)$.
Now I'll suppose that $k$ is of characteristic $0$.
The fact that $E^{n-k}:\bigwedge^k V\to\bigwedge^{2n-k}V$ is a bijection then follows from the fact that for any irreducible representation $W$ of $sl_2$, and for any weight $m\in\mathbb N$ (in our case $m=n-k$), $E^m:W_{-m}\to W_m$ is bijective ($W_N\subset W$ is the space of vectors of weight $N$, and it is $1$ or $0$ dimensional).
