On the Chern-Ricci form of a twisted almost Kähler structure

Let $(M,g,J,\omega )$ be an almost Kähler manifold. For any smooth function $f$ on $M$, one can associate an automorphism $\psi \in Aut(TM)$ for which the Kähler form is invariant. Using $\psi$, one can ``twist" the metric $g$ and almost complex structure $J$ to obtain a new almost Kähler structure $(g^{\psi },J^{\psi },\omega )$ on $M$. Let $\tilde{D}$ denote the Chern connection of $(g^{\psi },J^{\psi },\omega )$ and let $K^{-1}$ denote the anti-canonical bundle of $(TM,J^{\psi })$. In the current paper, we give an explicit formula for the local connection 1-form $\alpha$ associated to the pair $(K^{-1},\tilde{D})$. The Chern-Ricci form of $(g^{\psi },J^{\psi },\omega )$ is then ${\rho }_{\tilde{D}}=-d\alpha$. We note that under certain conditions the aforementioned formula assumes a simpler form when applied to the calculation of $\alpha$. We illustrate this with some examples.