The Weil–Petersson current on Douady spaces

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