Holomorphic Jet Modules and Holomorphic Connections for Noncommutative Complex Curves

TL;DR

This paper extends Atiyah's holomorphic jet bundle formalism to noncommutative complex curves, constructing canonical holomorphic structures and connections on jet modules, and characterizing their existence via the Atiyah class, with applications to quantum projective lines.

Contribution

It introduces a noncommutative analogue of Atiyah's correspondence for holomorphic vector bundles and connections on noncommutative complex curves, including quantum projective lines.

Findings

Constructed a canonical holomorphic structure on jet modules over noncommutative curves.

Proved that the existence of a holomorphic connection is equivalent to the splitting of the jet sequence.

Applied the theory to quantum projective lines to determine bimodule connection conditions.

Abstract

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 $(E,\overline{\nabla}_E)$ over such an algebra $\mathcal{A}$, we construct a canonical holomorphic structure $\overline{\nabla}_J$ on the first jet module $J_E^1\,$, making the jet sequence \[ 0\longrightarrow \Omega^{1,0}(\mathcal{A})\otimes_{\mathcal A}E\longrightarrow J_E^1\longrightarrow E\longrightarrow 0 \] exact in the holomorphic category. The association $(E,\overline\nabla_E)\rightsquigarrow(J_E^1\,,\overline\nabla_J)$ 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 $\mathbb{CP}_q^1\,$ and determine when $\overline{\nabla}_J$ defines a bimodule connection, assuming that $\overline{\nabla}_E$ does.