Local comparison theorems for Kähler Manifolds

We establish a sharp relative volume comparison theorem for small balls on Kähler manifolds with lower bound on Ricci curvature, assuming real analyticity of the etric. The model spaces being compared to are complex space forms, that is, Kähler manifolds with constant holomorphic sectional curvature. Moreover, we give an example showing that on Kähler manifolds, the pointwise Laplacian comparison theorem does not hold when the Ricci curvature is bounded from below.