Noncommutative complex geometry of the quantum projective space

TL;DR

This paper develops a noncommutative complex geometric framework for quantum projective spaces, defining holomorphic structures, identifying sections, and verifying classical theorems like Riemann-Roch and Serre duality in the quantum setting.

Contribution

It introduces holomorphic structures on quantum projective spaces, characterizes their sections, and establishes quantum analogs of classical geometric theorems.

Findings

Holomorphic structures on quantum projective spaces defined

Quantum homogeneous coordinate ring identified

Riemann-Roch and Serre duality verified for specific cases

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 $\qp^{2}_q$.