NEW POLARIZATIONS ON THE MODULI SPACES AND THE THURSTON COMPACTIFICATION OF TEICHMULLER SPACE

Given a foliation F with closed leaves and with certain kinds of singu larities on an oriented closed surface Sigma, we construct in this pap er an isotropic foliation on M(Sigma), the moduli space of flat G-conn ections, for G any compact simple simply connected Lie-group. We descr ibe the infinitesimal structure of this isotropic foliation in terms o f the basic cohomology with twisted coefficients of F. For any pair (F , g), where g is a singular metric on Sigma compatible with F, we cons truct a new polarization on the symplectic manifold M'(Sigma), the ope n dense subset of smooth points of M(Sigma). We construct a sequence o f complex structures on Sigma, such that the corresponding complex str uctures on M'(Sigma) converges to the polarization associated to (F, g ). In particular we see that the Jeffrey-Weitzman polarization on the SU(2)-moduli space is the limit of a sequence of complex structures in duced from a degenerating family of complex structures on Sigma, which converges to a point in the Thurston boundary of Teichmuller space of Sigma. As a corollary of the above constructions, we establish a cert ain discontinuiuty at the Thurston boundary of Teichmiiller space for the map from Teichmuller space to the space of polarizations on M'(Sig ma). For any reducible finite order diffeomorphism of the surface, our constuction produces an invariant polarization on the moduli space.