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Answer to a question by Fujita on variation of Hodge structures. (English) Zbl 1388.14037

Oguiso, Keiji (ed.) et al., Higher dimensional algebraic geometry. In honour of Professor Yujiro Kawamata’s sixtieth birthday. Proceedings of the conference, Tokyo, Japan, January 7–11, 2013. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-86497-046-4/hbk). Advanced Studies in Pure Mathematics 74, 73-102 (2017).
Summary: We first provide details for the proof of Fujita’s second theorem for Kähler fibre spaces over a curve, asserting that the direct image \(V\) of the relative dualizing sheaf splits as the direct sum \(V= A\oplus Q\), where \(A\) is ample and \(Q\) is unitary flat.
Our main result then answers in the negative the question posed by Fujita whether \(V\) is semi-ample. In fact, \(V\) is semi-ample if and only if \(Q\) is associated to a representation of the fundamental group of \(B\) having finite image. Our examples are based on hypergeometric integrals.
For the entire collection see [Zbl 1388.14012].

MSC:

14D07 Variation of Hodge structures (algebro-geometric aspects)
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
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