New polarizations on the moduli spaces and the Thurston compactification of Teichmüller space

Given a foliation F with closed leaves and with certain kinds of singularities on an oriented closed surface Σ, we construct in this paper an isotropic foliation on M(Σ), the moduli space of flat G-connections, for C any compact simple simply connected Lie-group. We describe the infinitesimal structure of this isotropic foliation in terms of the basic cohomology with twisted coefficients of F. For any pair (F, g), where g is a singular metric on Σ compatible with F, we construct a new polarization on the symplectic manifold M′(Σ), the open dense subset of smooth points of M(Σ). We construct a sequence of complex structures on Σ, such that the corresponding complex structures on M′(Σ) 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 induced from a degenerating family of complex structures on Σ, which converges to a point in the Thurston boundary of Teichmüller space of Σ. As a corollary of the above constructions, we establish a certain discontinuity at the Thurston boundary of Teichmüller space for the map from Teichmüller space to the space of polarizations on M′(Σ). For any reducible finite order diffeomorphism of the surface, our constuction produces an invariant polarization on the moduli space.