The PhD thesis of Dan Freed contains the computation of the Ricci curvature of an infinite-dimensional Kähler manifold.  I don't have a scanned copy of the thesis with me, but it's probably condensed in his 1988 paper in the Journal of Differential Geometry: [The Geometry of Loop Groups][1].


  [1]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jdg/1214442279