Local volume comparison for Kahler manifolds

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