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Positive definiteness conjecture for Heckenberger–Kolb Kähler structures

Let G/LSG/L_S be an irreducible quantum flag manifold, let Oq(G/LS)\mathcal{O}_q(G/L_S) be its coordinate algebra, and let Ωq∙(G/LS)\Omega_q^\bullet(G/L_S) be the Heckenberger–Kolb calculus. The associated covariant complex structure is denoted by Ω(∙,∙)\Omega^{(\bullet,\bullet)}, and let κ\kappa be its real, left-invariant (1,1)(1,1)-form, unique up to real scalar multiple.

Positive definiteness conjecture. For q∈R∖{−1,0}q\in\mathbb{R}\setminus\{-1,0\}, the pair (Ω(∙,∙),κ)(\Omega^{(\bullet,\bullet)},\kappa) is a positive definite covariant Kähler structure.

This conjecture extends the corresponding result for quantum projective space, where positivity is known for all positive qq sufficiently close to 11. It was originally proposed in the cited work, while the assertion for general irreducible quantum flag manifolds remains open.

Sources & referencesView supporting material

Primary source

Biswarup Das, Réamonn Ó Buachalla and Petr Somberg, “Compact Quantum Homogeneous Kähler Spaces”, arXiv:1910.14007 (2026).

noncommutative-kahler-geometryquantum-flag-manifoldsquantum-geometry

Progress summary

Refreshed
Open

The conjecture remains open: positivity is known near the classical parameter value, but no public result covers every allowed real parameter.

The conjecture asserts that the Heckenberger–Kolb complex structure and its invariant form define a positive Kähler structure on every irreducible quantum flag manifold for q∈R∖{−1,0}q\in\mathbb{R}\setminus\{-1,0\}. The relevant 2019 literature establishes the Kähler structure under broad conditions but explicitly leaves full positivity unresolved.

Known results

  • Kähler structures exist except possibly at finitely many values of qq; positivity was left for further study (2019).
  • Positivity holds for q>0q>0 in a sufficiently small interval around 11 (2019).
  • For every irreducible quantum flag manifold, the associated metric is positive definite on an open interval around q=1q=1 (2022).
  • Later work on quantum quadrics repeats these local and finite-exception results without extending them to all allowed qq (2022).

Current status (as of August 2026): the conjecture is open; positivity is established only for qq sufficiently close to 11, while the assertion for all q∈R∖{−1,0}q\in\mathbb{R}\setminus\{-1,0\} remains unsettled.

Sources

Solutions 0

No solutions have been posted yet.