Noncommutative complex geometry of the quantum projective space

Abstract

We define holomorphic structures on canonical line bundles of the quantum projective space \qp^{\ell}_q 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 \qp^{\ell}_q is naturally presented by a twisted positive Hochschild cocycle. Finally, we verify the main statements of Riemann-Roch formula and Serre duality for \qp^{1}_q and \qp^{2}_q