Noncommutative Kähler structures on quantum homogeneous spaces

Kähler structures on quantum irreducible flag manifolds

“…This proves the conjecture formulated in [ÓBu17,Conjecture 4.25], with the possible exception of finitely many values of q. Since for the quantum projective spaces this result is valid for all 0 < q < 1, we expect this to be the case in full generality.…”

“…The first result is Theorem 4.2, which shows that the Heckenberger-Kolb calculus Ω • over C q [G/P S ] becomes a differential * -calculus in a natural way. The second result is Theorem 5.9, which shows the existence of a Kähler structure on Ω • , thus proving the conjecture formulated in [ÓBu17,Conjecture 4.25].…”

“…(3) It is a general fact that every left C q [G]-coinvariant d-closed form is central, see [ÓBu17,Corollary 4.6] (this result requires B + Ω 1 = Ω 1 B + , which holds in our case as explained in the next subsection). Since κ is left C q [G]-coinvariant by Lemma 5.4 and d-closed, we conclude that it is central.…”

“…In this section we recall various notions related to the description of differential calculi on quantum spaces. In particular we consider the notions of Hermitian and Kähler structures, as defined in [ÓBu17]. Moreover we recall some aspects of Takeuchi's categorical equivalence, which is a quite useful tool when dealing with differential calculi.…”

Twisted Hochschild homology of quantum flag manifolds: 2-cycles from invariant projections

“…Indeed, classically the flag manifolds U/K S are Kähler and hence admit these two-forms with special properties. The study of analogues of Kähler forms in the quantum setting is important from the point of view of non-commutative complex and Kähler geometry, see for instance [BeSm13,ÓBu16,ÓBu17].…”