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".