The Weil–Petersson current on Douady spaces

The Douady space of compact subvarieties a Kähler manifold is equipped with the Weil–Petersson current, which everywhere positive local continuous potentials, and class C ∞ when restricted to locus smooth fibers. There Quillen metric known exist, whose Chern form equal form. In algebraic case, we show that can be extended determinant line bundle as singular hermitian metric. On other hand in such way yields current. We general theorem comparing holomorphic bundles metrics are isomorphic over complement snc divisor B. They differ by arising from flat bundle. forms current integration support B further related its normal latter zero case spaces due Yoshikawa on for families curves.