 Solution

Take the threshold $k_0\ge 1$ in the stated spectral gap for powers of a positive line bundle. For $q\ge 1$ and $k\ge k_0$, it givesA 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 $\boxed{H^q(X,L^k)=0(q\ge 1,k\ge k_0)}$. 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.