Kähler geometry of Douady spaces

We consider the generalized Petersson–Weil metric on the moduli space of compact submanifolds of a Kähler manifold or a projective variety. It is extended as a positive current to the space of points corresponding to reduced fibers, and estimates are shown. For moduli of projective varieties the Petersson–Weil form is the curvature of a certain determinant line bundle equipped with a Quillen metric. We investigate its extension to the compactified moduli space.