Positive line modules over the irreducible quantum flag manifolds

Autor: Fredy Díaz García, Andrey O. Krutov, Réamonn Ó Buachalla, Petr Somberg, Karen R. Strung
Rok vydání: 2022
Předmět:
Zdroj: Letters in Mathematical Physics. 112
ISSN: 1573-0530
0377-9017
DOI: 10.1007/s11005-022-01619-x
Popis: 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 $\mathcal{O}_q(G/L_S)$. 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 K\"ahler 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.
Comment: 30 pages, 2 tables. Minor changes. To appear in Letters in Mathematical Physics
Databáze: OpenAIRE
Externí odkaz: