Spectral gap for powers of a positive line bundle
= Spectral gap for powers of a positive line bundle
{title2=$\Delta^{0,q}_{L^k}\geq\varepsilon k\quad(q\geq1)$}
For a positive Hermitian <holomorphic line bundle> on compact $X$, using its curvature as the <Kähler form>, a linear lower bound in $k$ for the full <Dolbeault Laplacian> on positive antiholomorphic degrees rules out harmonic forms for sufficiently large $k$. The <Dolbeault theorem> and <Dolbeault Hodge decomposition> then imply $H^{q>0}(X,L^k)=0$.