# Relation between a hermitian inner product and a complex symplectic form

A complex symplectic space is a pair $(V, \omega)$, where $V$ is a complex vector space of complex dimension $2n$ and $\omega$ is a non-degenerate skew-symmetric $\mathbb{C}$-bilinear form. My question is:

Can we define a Hermitian inner product $H(-,-)$ on $V$ out of the complex symplectic form $\omega(-,-)$ in a similar fashion to case of a real symplectic space $W$ together with a compatible complex structure $J: W\rightarrow W$?

• Have you tried any examples? – Ted Shifrin Oct 19 '17 at 0:36
• I tried to cook up different hermitian inner products from the complex $\omega$. All of them failed to be hermitian, because the symplectic form is not sesqui-linear, but $\mathbb{C}$-bilinear. My only working idea so far is to think of V as a real $4n$-dimensional symplectic space with a complex structure $J: V\rightarrow V$ equipped with a real symplectic form $\omega_{\mathbb{R}}$. Then my hermitian inner product would be: $H(x, y)=\omega_{\mathbb{R}}(x,Jy)+i\omega_{\mathbb{R}}(x, y)$ – Flavius Aetius Oct 19 '17 at 9:36