Spectral gap for powers of a positive line bundle (${\Delta }_{L^k}^{0,q}\ge \epsilon k(q\ge 1,k\gg 1)$)

Let $(L,h)$ be a positive holomorphic line bundle on a compact complex manifold $X$ of complex dimension $n$, and use $\omega =i{F}_L$ 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 $q\ge 1$,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 $ik{F}_L=k\omega$ 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.^[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.