TITLE: moduli spaces are kahler?
QUESTION [15 upvotes]: I often heard from experts that "moduli spaces are Kahler". This sounds as a meta-theorem asserting that every time one defines reasonable moduli spaces, then there is a standard strategy to see (proof) if that space is Kahler or not.
I'm mainly interested in moduli spaces of geometric structures on surfaces: Teichmuller (ok, that's I know is Kahler), Moduli space of complex projective structures (is this space kahler? where can I find a reference?), et cetera...
Is there a standard (or not) reference for this kind of problems?

REPLY [2 votes]: The moduli space of holomorphic normal projective connections is an affine space, or empty, as it is identified with the collection of all holomorphic 1-cocycles whose coboundary is a suitable ``traceless Atiyah class'' of the tangent bundle. The paper
Robert Molzon and Karen Pinney Mortensen, 
The Schwarzian derivative for maps between manifolds with complex projective connections, Trans. Amer. Math. Soc. 348 (1996), no. 8, 3015–3036. MR 1348154 (96j:32028) 27, 55
is perhaps the best introduction to the theory of complex projective connections. I don't know a good reference for the relation to the Atiyah class, though it appears somewhere in the work of Kobayashi. The moduli space of flat holomorphic projective connections on a compact complex manifold is a complex subvariety, not known to be smooth.