# The Weil-Petersson current on Douady spaces

## TL;DR

This paper studies the Weil-Petersson current on Douady spaces of compact subvarieties of Kähler manifolds, extending the Quillen metric as a singular hermitian metric and comparing line bundles with singular metrics.

## Contribution

It extends the Quillen metric to the determinant line bundle as a singular hermitian metric and compares holomorphic line bundles with singular hermitian metrics over divisors.

## Key findings

- The Weil-Petersson current is positive with continuous potentials.
- The Quillen metric extends to the determinant line bundle as a singular hermitian metric.
- Differences in line bundles with singular metrics are characterized by divisors and flat bundles.

## Abstract

The Douady space of compact subvarieties of a K\"ahler manifold is equipped with the Weil-Petersson current, which is everywhere positive with local continuous potentials, and of class $C^\infty$ 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.