A complete or open Káhler manifold with positive definite Ricci tensor is simply connected? is there any counterexample?
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Let $S\subset \mathbb{P}^1$ denote a finite subset consisting of $n\geq 2$ points. The FubiniStudy metric on $\mathbb{P}^1$ induces a Kahler metric on $X= \mathbb{P}^1\backslash S$ with a positivedefinite Ricci tensor. $X$ is open and Kahler, but $\pi_1(X, *)$ is free on $n1$ generators. Maybe you want to take the manifold to be complete and noncompact? As you probably know, it is a theorem of Kobayashi that a compact Kahler manifold with positive definite Ricci tensor is simply connected. 

