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Green's functions and complex Monge-Ampère equations

Uniform $L^1$ and lower bounds are obtained for the Green's function on compact Kähler 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 Kähler 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-Ampère 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-Ampère equations with a sharper dependence on the function on the right hand side.