Positive Line Bundles Over the Irreducible Quantum Flag Manifolds
Fredy D\'iaz Garc\'ia, Andrey Krutov, R\'eamonn \'O Buachalla, Petr, Somberg, Karen R. Strung

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 . Building on the recently established noncommutative Borel-Weil theorem, every covariant line bundle over can be identified as positive, negative, or flat, and…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Algebra and Geometry · Advanced Topics in Algebra
