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The Weil-Petersson current on Douady spaces

Reynir Axelsson, Georg Schumacher

Abstract

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.

The Weil-Petersson current on Douady spaces

Abstract

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 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 . 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 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.

Paper Structure

This paper contains 14 sections, 23 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Infinitesimal theory of Douady spaces -- Notions
  3. Geodesic curvature and Weil-Pe-ters-son form
  4. The Weil-Pe-ters-son current
  5. Determinant Line Bundle
  6. BGS-Theorem and Bott-Chern form
  7. Barlet class
  8. Yoshikawa's result
  9. Application of the BGS-Theorem
  10. Application of Yoshikawa's Theorem
  11. Smooth base curves
  12. Singular base curves
  13. Singular hermitian metrics and residues
  14. Application to Douady spaces

Key Result

Theorem 1

Let $Z$ be a complex manifold and $B\subset Z$ a smooth hypersurface. Let $({\mathrsfs L} ,h)$ be a holomorphic line bundle on $Z$, whose restriction to $Z'=Z\setminus B$ is isomorphic to the trivial line bundle $({\mathrsfs O} _{Z'},1)$ carrying the trivial hermitian metric. Then where $\beta$ is a rational number, and $L$ a flat line bundle. There exists a holomorphic section $v$ of the normal

Theorems & Definitions (36)

  • Theorem
  • Theorem
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.5: sch:inv12b-sa-s
  • Theorem 3.6: Fujiki
  • Theorem 4.1: v2
  • Proposition 4.2: cf. a-s
  • ...and 26 more