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 $C^{\infty }$ 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.

中文翻译：

---

杜阿迪空间上的Weil–Petersson流

Kähler流形的紧致子集的Douady空间配备了Weil–Petersson电流，该电流在任何地方都具有局部连续势，并且是正的 $C^{\infty }$仅限于光滑纤维的位置。已知存在Quillen度量，其Chern形式等于Weil-Petersson形式。在代数情况下，我们表明Quillen度量可以作为奇异的Hermitian度量扩展到行列式束。另一方面，行列式束可以扩展，以使Quillen度量产生奇异的Hermitian度量，其Chern形式等于Weil-Petersson电流。我们展示了一个一般性定理，比较配备了奇异Hermitian度量的全纯线束，它们在snc除数B的补码上是同构的。它们的区别在于因除数产生的线束和扁平线束。陈氏形式的不同之处在于集成B中支持的最新趋势以及与其正常捆绑相关的其他电流。在双数空间的情况下，后者的电流等于零，这是由于吉川关于奇异族在曲线上的Quillen度量的一个定理。