Positive Line Bundles Over the Irreducible Quantum Flag Manifolds
arXiv:1912.08802 · doi:10.1007/s11005-022-01619-x
Abstract
Noncommutative Kähler structures were recently introduced by the second author 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 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 . Building on the recently established noncommutative Borel-Weil theorem, every covariant line bundle over can be identified as positive, negative, or flat, and hence we can conclude that each Kähler structure is of Fano type. Moreover, it proves possible to extend the Borel-Weil theorem for to a direct noncommutative generalisation of the classical Bott-Borel-Weil theorem for positive line bundles.
arXiv comment30 pages, 2 tables. Minor changes. To appear in Letters in Mathematical Physics
References in corpus (11)
- Connections on central bimodules
- CQG algebras: a direct algebraic approach to compact quantum groups
- Categories of holomorphic vector bundles on noncommutative two-tori
- On Curvature in Noncommutative Geometry
- Linear connections on the quantum plane
- Dirac Operators on Quantum Projective Spaces
- Anti-selfdual Connections on the Quantum Projective Plane: Monopoles
- Differential forms via the Bernstein-Gelfand-Gelfand resolution for quantized irreducible flag manifolds
- Kähler structures on quantum irreducible flag manifolds
- Holomorphic Relative Hopf Modules over the Irreducible Quantum Flag Manifolds
- Anti-selfdual Connections On The Quantum Projective Plane: Instantons