Karen Strung
(Institute of Mathematics, Czech Academy of Sciences)
May 12, 2021
Organized by Europe GNCG Seminar.

Positive Line Bundles Over the Irreducible Quantum Flag Manifolds

Noncommutative Kähler structures were recently introduced by Ó Buachalla as a framework for studying noncommutative Kähler geometry on quantum homogeneous spaces. The notion of a positive vector bundle directly generalises to this setting. For covariant Kähler 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 O_q(G/L_S). Building on the recently established noncommutative Borel-Weil theorem, every covariant line bundle over O_q(G/L_S) can be identified as positive, negative, or flat, and hence we can conclude that each Kähler structure is of Fano type. This is joint work with Díaz García, Ó Buachalla, Krutov, and Somberg.

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