Let G be a connected complex semisimple Lie group. Let Gamma be a cocompactlattice in G. Matsushima raised the question whether the canonical complex structure on X = Gamma/G is locally rigid. Raghunathan showed that whenever G has no three dimensional components, the canonical complex structure on Gamma/G is locally rigid. In this paper, we show that when G is SL(sub 2)(C), nontrivial deformations of the canonical complex structure on X exist if and only if the first Betti number of the lattice Gamma is non-zero. We also show that the holomorphic representations rho of G not occurring in H(sup i)(X,O) for any i greater than or equal to 0 are precisely those for which H(sup i)(Gamma,rho) = (0) for all i greater than or equal to 0.
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