The Levi-Civita connection and Chern connections for cocycle deformations of Kähler manifolds

We consider unitary cocycle deformations of covariant -differential calculi. We prove that complex structures, holomorphic bimodules and Chern connections on the deformed calculus are twists of their untwisted counterparts. Moreover, for cocycle deformations of a class of classical Kähler manifolds, the Levi-Civita connection on the space of one forms of the deformed calculus is shown to be a direct sum of the Chern connections on the twisted holomorphic and the anti-holomorphic bimodules. Our class of examples also include cocycle deformations of the Heckenberger-Kolb calculi.