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We show that tensoring the Laplace and Dolbeault-Dirac operators of aK\"ahler structure (with closed integral) by a negative Hermitian holomorphicmodule, produces operators with spectral gaps around zero. The proof is basedon the recently established Akizuki-Nakano identity of a noncommutativeK\"ahler structure. This general framework is then applied to theHeckenberger-Kolb calculi of the irreducible quantum flag manifolds, and it isshown that twisting their Dirac and Laplace operators by negative line bundlesproduces a spectral gap, for q sufficiently close to 1. The main technicalchallenge in applying the framework is to establish positivity of the quantumFubini-Study metric of the quantum flag manifold. Importantly, combiningpositivity with the noncommutative hard Lefschetz theorem, it is additionallyobserved that the even degree de Rham cohomology groups of theHeckenberger-Kolb calculi do not vanish.

Spectral gaps for twisted Dolbeault-Dirac operators over the irreducible quantum flag manifolds