Positive Line Bundles Over the Irreducible Quantum Flag Manifolds

TL;DR

This paper develops cohomological criteria for positivity of line bundles on irreducible quantum flag manifolds within noncommutative K"ahler geometry, extending classical theorems and classifying line bundles as positive, negative, or flat.

Contribution

It introduces simple cohomological criteria for positivity in noncommutative K"ahler structures and extends the Borel-Weil theorem to quantum flag manifolds.

Findings

All covariant line bundles over irreducible quantum flag manifolds are classified as positive, negative, or flat.

Every K"ahler structure on these manifolds is of Fano type.

A noncommutative generalisation of the Bott-Borel-Weil theorem is established.

Abstract

Noncommutative K\"ahler structures were recently introduced by the second author as a framework for studying noncommutative K\"ahler geometry on quantum homogeneous spaces. It was subsequently observed that the notion of a positive vector bundle directly generalises to this setting, as does the Kodaira vanishing theorem. In this paper, by restricting to covariant K\"ahler structures of irreducible type (those having an irreducible space of holomorphic one-forms) we provide simple cohomological criteria for positivity, offering a means to avoid explicit curvature calculations. These general results are applied to our motivating family of examples, the irreducible quantum flag manifolds $\mathcal{O}_q(G/L_S)$. Building on the recently established noncommutative Borel-Weil theorem, every covariant line bundle over $\mathcal{O}_q(G/L_S)$ can be identified as positive, negative, or flat, and hence we can conclude that each K\"ahler structure is of Fano type. Moreover, it proves possible to extend the Borel-Weil theorem for $\mathcal{O}_q(G/L_S)$ to a direct noncommutative generalisation of the classical Bott-Borel-Weil theorem for positive line bundles.