Positive line modules over the irreducible quantum flag manifolds

Noncommutative Kähler structures were recently introduced as a framework for studying noncommutative Kähler 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 Kähler structures of irreducible type (those having an irreducible space of holomorphic 1-forms) we provide simple cohomological criteria for positivity, allowing one to avoid explicit curvature calculations. These general results are applied to our motivating family of examples, the irreducible quantum flag manifolds O q ( G / L S ) . Building on the recently established noncommutative Borel–Weil theorem, every relative line module over O q ( G / L S ) can be identified as positive, negative, or flat, and it is then concluded that each Kähler structure is of Fano type.