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

Abstract: We consider unitary cocycle deformations of covariant $\ast$-differential calculi. We prove that complex structures, holomorphic bimodules and Chern connections can be deformed to their noncommutative counterparts under such deformations. If we start with a K\"ahler manifold, then the Levi-Civita connection on the space of one forms of the deformed calculus can be expressed as a direct sum of the Chern connections on the twisted holomorphic and the anti-holomorphic bimodules. Our class of examples include toric deformations considered by Mesland and Rennie as well as cocycle deformations of the Heckenberger-Kolb calculi.