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I am wondering if it is known whether the unitary group $U(n)$ is a Kahler manifold, and, if so, what is a reference for this.

Let $G$ be a Lie group acting smoothly and freely on a smooth manifold $M$. Suppose that the quotient space $M/G$ is a topological manifold. Do we have $$\dim(M/G)=\dim M-\dim G?$$ Notes: This ...