On the Chern-Ricci form of a twisted almost K\"{a}hler structure

Abstract

Let $(M,g,J,\omega)$ be an almost K\"{a}hler manifold. For any smooth function $f$ on $M$, one can associate an automorphism $\psi\in \mbox{Aut}(TM)$ for which the K\"{a}hler form is invariant. Using $\psi$, one can ``twist" the metric $g$ and almost complex structure $J$ to obtain a new almost K\"{a}hler structure $(g^\psi,J^\psi,\omega)$ on $M$. Let $\widetilde{D}$ 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 $(K^{-1},\widetilde{D})$. The Chern-Ricci form of $(g^\psi,J^\psi,\omega)$ is then $\rho_{\widetilde{D}}=-d\alpha$. 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.