The Weil-Petersson current on Douady spaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866911763996344320 |
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| author | Axelsson, Reynir Schumacher, Georg |
| author_facet | Axelsson, Reynir Schumacher, Georg |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1812_07623 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | The Weil-Petersson current on Douady spaces Axelsson, Reynir Schumacher, Georg Complex Variables 32G10, 35G55 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. |
| title | The Weil-Petersson current on Douady spaces |
| topic | Complex Variables 32G10, 35G55 |
| url | https://arxiv.org/abs/1812.07623 |