### Abstract

For a holomorphic family of simple Hermitian-Einstein holomorphic vector bundles over a compact Kähler manifold, the locally free part of the associated direct image sheaf over the parameter space forms a holomorphic vector bundle, and it is endowed with a Hermitian metric given by the L^{2} pairing using the Hermitian-Einstein metrics. Our main result in this paper is to compute the curvature of the L^{2}-metric. In the case of a family of Hermitian holomorphic line bundles with fixed positive first Chern form and under certain curvature conditions, we show that the L^{2}-metric is conformally equivalent to a Hermitian-Einstein metric. As applications, this proves the semi-stability of certain Picard bundles, and it leads to an alternative proof of a theorem of Kempf.

Original language | English |
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Pages (from-to) | 649-661 |

Number of pages | 13 |

Journal | American Journal of Mathematics |

Volume | 120 |

Issue number | 3 |

Publication status | Published - Jun 1998 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

^{2}-metric on the direct image of a family of Hermitian-Einstein vector bundles.

*American Journal of Mathematics*,

*120*(3), 649-661.