  We prove certain gradient and eigenvalue estimates, as well as the heat
kernel estimates, for the Hodge Laplacian on $(m,0)$ forms, i.e., sections of
the canonical bundle of K\"ahler manifolds, where $m$ is the complex dimension
of the manifold. Instead of the usual dependence on curvature tensor, our
condition depends only on the Ricci curvature bound. The proof is based on a
new Bochner type formula for the gradient of $(m, 0)$ forms, which involves
only the Ricci curvature and the gradient of the scalar curvature.
