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I wanted to find the proof showing that the space of Kähler metrics on a given Kähler manifold $M$ is simply connected.

Any reference containing such topological results on the space of Kähler metrics would be highly appreciated.

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That space is contractible, thus simply connected:

Let $h_1$ be any Kahler metric on the given manifold, and let $h_2$ be any other Kahler metric. We claim the line segment $l(t) = h_2 + t (h_1 - h_2)$ is in the space of Kahler metrics for $0 \leq t \leq 1$. Any point on it is clearly $d$-closed. For any fixed nonzero tangent vector $\xi$, the function $t \mapsto l(t)(\xi, \xi)$ is affine (a linear function plus a constant), thus monotone. But it is positive at both $t = 0$ and $t = 1$, so it is positive for all $t$ in between them. Thus all points on the line segment are positive definite, that is, they are Kahler metrics.

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