Noncommutative complex geometry of the quantum projective space

We define holomorphic structures on canonical line bundles of the quantum projective space CPqℓ and identify their space of holomorphic sections. This determines the quantum homogeneous coordinate ring of the quantum projective space. We show that the fundamental class of CPqℓ is naturally presented by a twisted positive Hochschild cocycle. Finally, we verify the main statements of the Riemann–Roch formula and the Serre duality for CPq1 and CPq2.

► We define holomorphic structures on canonical line bundles of the quantum projective space.
► We identify the space of holomorphic sections of these line bundles.
► This determines the quantum homogeneous coordinate ring of the quantum projective space.
► The fundamental class of the quantum projective space is represented by a twisted positive Hochschild cocycle.
► We verify the main statements of the Riemann–Roch formula and the Serre duality for the quantum projective plane.