Geometry of moduli spaces of Higgs bundles

We construct a Petersson-Weil type K\"ahler form on the moduli spaces of Higgs bundles over a compact K\"ahler manifold. A fiber integral formula for this form is proved, from which it follows that the Petersson-Weil form is the curvature of a certain determinant line bundle, equipped with a Quillen metric, on the moduli space of Higgs bundles over a projective manifold. The curvature of the Petersson-Weil K\"ahler form is computed. We also show that, under certain assumptions, a moduli space of Higgs bundles supports of natural hyper-K\"ahler structure.