= Solution
Take the threshold $k_0\geq1$ in the stated <spectral gap for powers of a positive line bundle>. For $q\geq1$ and $k\geq k_0$, it gives
$$
\langle\Delta''_{L^k}\alpha,\alpha\rangle\geq\varepsilon k\|\alpha\|^2
\quad\text{on }\mathcal A^{0,q}(X,L^k).
$$
A harmonic form must therefore be zero. The <Dolbeault Hodge decomposition> and the <Dolbeault theorem> identify its zero harmonic space with $H^q(X,\mathcal O(L^k))$. Hence \b[$\boxed{H^q(X,L^k)=0\quad(q\geq1,\ k\geq k_0)}$]. For $q>\dim_{\mathbb C}X$ the form spaces already vanish. The gap controls the lowest eigenvalue including zero, so it rules out harmonic forms rather than just controlling the positive spectrum.
Back to article page
