Holomorphic structures on the quantum projective line

TL;DR

This paper explores the extension of complex structures from the classical 2-sphere to its quantum analogue, the quantum projective line, establishing algebraic and geometric parallels through Hochschild cocycles.

Contribution

It introduces a framework for understanding complex structures on quantum spheres using twisted Hochschild cocycles, linking classical and quantum geometric concepts.

Findings

Identification of the quantum homogeneous coordinate ring with the quantum plane

Construction of a twisted positive Hochschild cocycle for the quantum sphere

Extension of complex structure concepts to quantum geometries

Abstract

We show that much of the structure of the 2-sphere as a complex curve survives the q-deformation and has natural generalizations to the quantum 2-sphere - which, with additional structures, we identify with the quantum projective line. Notably among these is the identification of a quantum homogeneous coordinate ring with the coordinate ring of the quantum plane. In parallel with the fact that positive Hochschild cocycles on the algebra of smooth functions on a compact oriented 2-dimensional manifold encode the information for complex structures on the surface, we formulate a notion of twisted positivity for twisted Hochschild and cyclic cocycles and exhibit an explicit twisted positive Hochschild cocycle for the complex structure on the sphere.