## Title data

Catanese, Fabrizio ; Dettweiler, Michael:

**Answer to a question by Fujita on Variation of Hodge Structures.**

*In:* Oguiso, Keiji ; Birkar, Caucher ; Ishii, Shihoko ; Takayama, Shigeharu
(ed.):
Higher Dimensional Algebraic Geometry : In Honor of Professor Yujiro Kawamata's sixtieth Birthday. -
Tokyo
: Mathematical Society of Japan
,
2017
. - pp. 73-102
. - (Advanced Studies in Pure Mathematics
; 74
)

ISBN 978-4-86497-046-4

DOI: https://doi.org/10.2969/aspm/07410073

Review: |

## Project information

Project title: |
Project's official title Project's id DFG Forschergruppe 790 “Classification of algebraic surfaces and compact complex manifolds” No information DFG grant DE 1442/4-1 No information |
---|---|

Project financing: |
Deutsche Forschungsgemeinschaft |

## Abstract in another language

Let f:X→B be a fibration of a compact Kähler manifold X over a projective curve B, and let V denote the direct image f∗ωX/B of the relative dualizing sheaf. In the note [Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), no. 7, 183–184; MR0510945], T. Fujita announced that V splits as a direct sum V=A⊕Q, where A is an ample vector bundle and Q is a unitary flat bundle. He, however, only sketched the proof in [op. cit.]. One of the purposes of the present article is to provide the missing details concerning the proof. Another and main purpose of this paper is to investigate the question posed by Fujita in 1982 [in Classification of algebraic and analytic manifolds (Katata, 1982), 591–630, Progr. Math., 39, Birkhäuser Boston, Boston, MA, 1983; MR0728620] which asks whether or not the direct image V is semi-ample. The authors show that the question has a negative answer, by showing that there exist surfaces X of general type endowed with a fibration f:X→B such that V=A⊕Q1⊕Q2, where A is an ample vector bundle, and the flat unitary rank-two summands Q1,Q2 have infinite monodromy group, which implies in particular V is not semi-ample.

## Further data

Item Type: | Article in a book |
---|---|

Refereed: | Yes |

Keywords: | Variation of Hodge structure; relative dualizing sheaf; semiampleness |

Subject classification: | 2010 Mathematics Subject Classification: 14D07, 14C30, 32G20, 33C60 |

Institutions of the University: | Faculties Faculties > Faculty of Mathematics, Physics und Computer Science Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics IV (Number Theory) Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics IV (Number Theory) > Chair Mathematics IV (Number Theorie) - Univ.-Prof. Dr. Michael Dettweiler |

Result of work at the UBT: | Yes |

DDC Subjects: | 500 Science > 510 Mathematics |

Date Deposited: | 05 May 2023 10:44 |

Last Modified: | 05 May 2023 10:44 |

URI: | https://eref.uni-bayreuth.de/id/eprint/75962 |