• Corpus ID: 209405398

Positive Line Bundles Over the Irreducible Quantum Flag Manifolds

@article{Garcia2019PositiveLB,
  title={Positive Line Bundles Over the Irreducible Quantum Flag Manifolds},
  author={Fredy D'iaz Garc'ia and Andrey Krutov and R'eamonn 'O Buachalla and Petr Somberg and Karen R. Strung},
  journal={arXiv: Quantum Algebra},
  year={2019}
}

• Fredy D'iaz Garc'ia, Andrey Krutov, +2 authors Karen R. Strung
• Published 18 December 2019
• Mathematics
• arXiv: Quantum Algebra

Noncommutative Kahler structures were recently introduced by the third author as a framework for studying noncommutative Kahler 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 Kahler structures of irreducible type (those having an irreducible space of holomorphic $1$-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 Kahler 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.