# Kahler identities on almost Kahler manifolds

Suppose that $A$ is a unitary connection on a Hermittian differentiable vector bundle $E$ over a Kahler manifold $X$, then we have operators $$\bar{\partial}_A: \Omega_{X}^{p,q}(E)\to \Omega_{X}^{p,q+1}(E)$$ $${\partial}_A: \Omega_{X}^{p,q}(E)\to \Omega_{X}^{p+1,q}(E)$$ And we have the Kahler identities: $$\bar{\partial}_{A}^{*}=-i[\partial_A,\Lambda]$$ $${\partial}_{A}^{*}=i[\bar{\partial}_A,\Lambda]$$

Is this result also true for almost Kahler manifolds? Does anyone know a good reference for it? By definition, an almost Kahler manifold is a manifold endowed with $(g,J,\omega)$, where $g$ is a Riemannian metric, $\omega$ is a symplectic form, and $J$ is an almost complex structure, and they satisfy the compatibility condition $$g(X,Y)=\omega(X,JY)$$

• Could you elaborate on your assumptions on $E$ in each case? I assume that at the beginning $E$ is a hermitian holomorphic vector bundle, but in the desired setting, I don't think it makes sense to say $E$ is holomorphic if the base is not a complex manifold. Maybe I'm wrong. – Michael Albanese Apr 4 '15 at 18:36
• @MichaelAlbanese Thanks! It's already edited. – Boyu Zhang Apr 4 '15 at 19:47
• I think this is true. The essential input which justifies the calculation using standard coordinate is that $\nabla^{LC}\omega=d\omega=0$. – user44651 Jan 8 '16 at 5:12