Holomorphic Jet Modules and Holomorphic Connections for Noncommutative Complex Curves

We extend Atiyah’s holomorphic jet bundle formalism to holomorphic vector bundles over noncommutative algebras endowed with a bigraded differential calculus truncated at bidegree $(1,1)$; we refer to such structures as noncommutative complex curves. For a holomorphic vector bundle $(E,{\bar{\mathrm{\nabla }}}_E)$ over such an algebra $\mathcal{A}$, we construct a canonical holomorphic structure ${\bar{\mathrm{\nabla }}}_J$ on the first jet module $J_E^1$, making the jet sequence $$0⟶{\mathrm{\Omega }}^{1,0}(\mathcal{A}){\otimes }_{\mathcal{A}}E⟶J_E^1⟶E⟶0$$ exact in the holomorphic category. The association $(E,{\bar{\mathrm{\nabla }}}_E)⇝(J_E^1,{\bar{\mathrm{\nabla }}}_J)$ defines an endofunctor on the category of holomorphic vector bundles over $\mathcal{A}$. We define the notion of holomorphic connection in this setting and prove that a holomorphic vector bundle admits a holomorphic connection if and only if the jet sequence splits in the holomorphic category, or equivalently, if and only if its Atiyah class vanishes. This yields a noncommutative analogue of Atiyah’s classical correspondence for Riemann surfaces. Finally, we specialize to the quantum projective line ${\mathbb{C}\mathbb{P}}_q^1$ and determine when ${\bar{\mathrm{\nabla }}}_J$ defines a bimodule connection, assuming that ${\bar{\mathrm{\nabla }}}_E$ does.