Abstract
Noncommutative K & auml;hler structures were recently introduced as a framework for studying noncommutative K & auml;hler 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 & auml;hler 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 K & auml;hler structure is of Fano type.