The authors construct holomorphic jet modules over noncommutative complex curves and prove that a holomorphic vector bundle admits a holomorphic connection if and only if its Atiyah class vanishes, providing a noncommutative analogue of Atiyah's correspondence for Riemann surfaces.
Landi, Equivariant and holomorphic bundles on the quantum projective line
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Holomorphic Jet Modules and Holomorphic Connections for Noncommutative Complex Curves
The authors construct holomorphic jet modules over noncommutative complex curves and prove that a holomorphic vector bundle admits a holomorphic connection if and only if its Atiyah class vanishes, providing a noncommutative analogue of Atiyah's correspondence for Riemann surfaces.