  Uniform $L^1$ and lower bounds are obtained for the Green's function on
compact K\"ahler manifolds. Unlike in the classic theorem of Cheng-Li for
Riemannian manifolds, the lower bounds do not depend directly on the Ricci
curvature, but only on integral bounds for the volume form and certain of its
derivatives. In particular, a uniform lower bound for the Green's function on
K\"ahler manifolds is obtained which depends only on a lower bound for the
scalar curvature and on an $L^q$ norm for the volume form for some $q>1$. The
proof relies on auxiliary Monge-Amp\`ere equations, and is fundamentally
non-linear. The lower bounds for the Green's function imply in turn $C^1$ and
$C^2$ estimates for complex Monge-Amp\`ere equations with a sharper dependence
on the function on the right hand side.
