  We show that tensoring the Laplace and Dolbeault-Dirac operators of a
K\"ahler structure (with closed integral) by a negative Hermitian holomorphic
module, produces operators with spectral gaps around zero. The proof is based
on the recently established Akizuki-Nakano identity of a noncommutative
K\"ahler structure. This general framework is then applied to the
Heckenberger-Kolb calculi of the irreducible quantum flag manifolds, and it is
shown that twisting their Dirac and Laplace operators by negative line bundles
produces a spectral gap, for q sufficiently close to 1. The main technical
challenge in applying the framework is to establish positivity of the quantum
Fubini-Study metric of the quantum flag manifold. Importantly, combining
positivity with the noncommutative hard Lefschetz theorem, it is additionally
observed that the even degree de Rham cohomology groups of the
Heckenberger-Kolb calculi do not vanish.
