Answer to a question by Fujita on Variation of Hodge Structures

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Catanese, Fabrizio ; Dettweiler, Michael:
Answer to a question by Fujita on Variation of Hodge Structures.
In: Oguiso, Keiji ; Birkar, Caucher ; Ishii, Shihoko ; Takayama, Shigeharu (Hrsg.): Higher Dimensional Algebraic Geometry : In Honor of Professor Yujiro Kawamata's sixtieth Birthday. - Tokyo : Mathematical Society of Japan , 2017 . - S. 73-102 . - (Advanced Studies in Pure Mathematics ; 74 )
ISBN 978-4-86497-046-4
DOI: https://doi.org/10.2969/aspm/07410073

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Abstract

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.