= Spectral gap for powers of a positive line bundle
{title2=$\Delta^{0,q}_{L^k}\geq\varepsilon k\quad(q\geq1,\ k\gg1)$}
Let $(L,h)$ be a <positive holomorphic line bundle> on a compact <complex manifold> $X$ of complex <dimension> $n$, and use $\omega=iF_L$ as the <Kähler form>. There is a constant $C$ independent of the positive <integer> $k$ such that, on smooth $L^k$-valued $(0,q)$-forms with $q\ge1$,
$$
\langle\Delta''_{L^k}u,u\rangle_{L^2}
\ge(kq-C)\|u\|_{L^2}^2.
$$
In particular, for sufficiently large $k$, the <Dolbeault Laplacian> is bounded below by $k/2$ on every positive antiholomorphic degree.
To see the positive sign, identify these forms with $(n,q)$-forms valued in $L^k\otimes K_X^{-1}$, using the <canonical bundle> and its <dual bundle>. The <Bochner-Kodaira-Nakano identity> writes the quadratic form as a nonnegative <derivative> term plus the <curvature> commutator. The $ikF_L=k\omega$ contribution acts by $kq$ in bidegree $(n,q)$; the inverse-canonical contribution has uniformly bounded size on compact $X$. This gives $kq-C$. Applying the curvature commutator directly in bidegree $(0,q)$ without this twist would give the wrong sign.<Complex Analytic and Differential Geometry by Jean-Pierre Demailly>[Chapter VII, §§1–2, Bochner-Kodaira-Nakano identity and basic a priori inequality, printed pp.329–333]{ref}
The estimate rules out nonzero <harmonic forms> in these degrees. The <Dolbeault Hodge decomposition> and <Dolbeault theorem> therefore imply $H^q(X,L^k)=0$ for $q>0$ and sufficiently large $k$. The additive constant matters for small powers; the linear spectral gap is an eventual statement.
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