The homogeneous coordinate ring of the quantum projective plane

TL;DR

This paper develops a framework for holomorphic structures on line bundles over the quantum projective plane, linking these structures to the quantum homogeneous coordinate ring and Hochschild cocycles.

Contribution

It introduces holomorphic structures on line bundles in the quantum setting and connects them to the algebraic structure of the quantum projective plane.

Findings

Holomorphic sections define the quantum homogeneous coordinate ring.

Holomorphic structure is represented by a twisted positive Hochschild 4-cocycle.

Establishes a geometric-algebraic link in quantum geometry.

Abstract

We define holomorphic structures on canonical line bundles on the quantum projective plane. The space of holomorphic sections of these line bundles will determine the quantum homogeneous coordinate ring of $\qp^2_q$. We also show that the holomorphic structure of $\qp^2_q$ is naturally represented by a twisted positive Hochschild 4-cocycle.