Hopf cyclic cohomology in braided monoidal categories

We extend the formalism of Hopf cyclic cohomology to the context of braided categories. For a Hopf algebra in a braided monoidal abelian category we introduce the notion of stable anti-Yetter-Drinfeld module. We associate a para-cocyclic and a cocyclic object to a braided Hopf algebra endowed with a braided modular pair in involution in the sense of Connes and Moscovici. When the braiding is symmetric the full formalism of Hopf cyclic cohomology with coefficients can be extended to our categorical setting.Comment: 50 pages. One reference added. Proofs are visualized through braiding diagrams. Final version to appear in `Homology, Homotopy and Applications

Noncommutative complex geometry of the quantum projective space

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