moduli spaces are kahler? 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?
 A: The corresponding Kahler metric is called "Weil-Petersson metric"; it is often constructed using infinite-dimensional determinants of the corresponding Laplace operators. The standard reference is a sequence of papers 
by Bismut-Gillet-Soule: J.Bismut, H.Gillet and C.Soule, 
Analytic torsion and holomorphic determinant bundles, 
I,II,III, Comm. Math Phys., 115, 1988, 49-78, 79-126, 301-351; see also
http://arxiv.org/abs/math/0406408, http://arxiv.org/abs/math/0312172,
Leon A. Takhtajan, Lee-Peng Teo, "Weil-Petersson metric on the universal Teichmuller space I, II".
A: The Weil-Petersson metric is not complete metric in general but in the case of abelian varieties and $K3$ surfaces, the Weil-Petersson
metric turns out to be equal to the Bergman metric of the Hermitian symmetric
period domain, hence is in fact complete K\"ahler Einstein metric. Weil  and Ahlfors  showed that the Weil-Petersson metric is a Kahler metric and later Gang Tian gave a different proof for it also

There is a belief due to Fujiki-Tian saying that a moduli space with
  canonical metric (K\"ahler Einstein metric, cscK,...)is likely to be
  quasi-projective.

Recently Chi Li  http://arxiv.org/pdf/1502.06532.pdf prove that there is a canonical continuous Hermitian metric on
the Tian's CM-line bundle over the proper moduli space $M$ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson metric. Which Weil-Petersson metric is a Kahler metric
Let $\pi:X \to S$ be a projective family of canonically polarized varieties. Equip
the relative canonical bundle $K_{X/S}$ with the hermitian metric that is induced by the fiberwise K\"ahler-Einstein metrics. The log
Weil-Petersson form is equal, up to a numerical factor, to the fiber integral
$$\omega_{WP}=\int_{X_s}c_1\left(K_{X/S}\right)^{n+1}=\left(\int_{X_s}|A|_{\omega_s}^2\right)ds\wedge d\bar s$$
$A$ represents the Kodaira-Spencer class of the deformation,

There exists a Kahler metric on moduli space of Calabi-Yau varieties
  we call it again Weil Petersson metric and can be written as Ricci
  curvature of direct image of relative Line bundle.

Take holomorphic fiber space $\pi:X\to B$ and assume $\Psi_y$ be any local non-vanishing holomorphic section of Hermitian line bundle $\pi_*(K_{X/B}^l)$, then the Weil-Petersson (1,1)-form on a small ball $N_r(y)\subset B$ can be written as 
$$\omega_{WP}=-\sqrt{-1}\partial_y\bar{\partial_y}\log \left((\sqrt{-1})^{n^2}\int_{X_y}(\Psi_y\wedge \overline{\Psi_y})^{\frac{1}{l}}\right)$$
Note that $\omega_{WP}$ is globally defined on $B$

Fujiki-Schumacher considered the moduli space of Kahler manifolds
  admitting constant scalar curvature Kahler (cscK) metrics. They proved
  that the natural Weil-Petersson metric is always Kahler by
  interpreting it as the Chern curvature of a determinant line bundle
  equipped with Quillen metric.

A: 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. 
