Holomorphic structures on the quantum projective line

非标准全纯结构在量子射影线上的线丛上

Non-standard Holomorphic Structures on Line Bundles over the Quantum Projective Line

本文探讨量子射影线上的线丛非标准全纯结构，证明存在无限多个非规范等价结构，反驳了2011年提出的疑问。

量子射影线上的线丛的不动点与全纯结构

Fixed points and Holomorphic Structures on Line Bundles over the Quantum Projective Line

研究量子射影线上的线丛全纯结构不唯一性，通过不动点理论分析平坦$\bar\partial$连接，并将可逆解的存在性与特定非线性映射在适当Banach空间中的不动点存在性联系起来。

The homogeneous coordinate ring of the quantum projective plane

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.

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 $\qp^{2}_q$.

Differentiating the State Evaluation Map from Matrices to Functions on Projective Space

We show that the pure state evaluation map from $ M_{n}(\mathbb{C}) $ to $ C(\mathbb{C} \mathbb{P}^{n-1}) $ (a completely positive map of $ C^{*} $-algebras) extends to a cochain map from the universal calculus on $ M_{n}(\mathbb{C} ) $ to the holomorphic $ \bar{\partial} $ calculus on $ \mathbb{C} \mathbb{P}^{n-1} $. The method uses connections on Hilbert $ C^{*} $-bimodules. This implies the existence of various functors, including one from $ M_{n}(\mathbb{C}) $ modules to holomorphic bundles on $ \mathbb{C} \mathbb{P}^{n-1} $.