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

Abstract

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