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On birationally trivial families and adjoint quadrics

Part of: Birational geometry Surfaces and higher-dimensional varieties Families, fibrations Cycles and subschemes

Published online by Cambridge University Press:  04 July 2022

Luca Cesarano ,
Luca Rizzi Open the ORCID record for Luca Rizzi [Opens in a new window]  and
Francesco Zucconi Open the ORCID record for Francesco Zucconi [Opens in a new window]
Luca Cesarano
Affiliation:
Mathematisches Institut, Universität Bayreuth, Universitätsstraße 30, 95447 Bayreuth, Germany (luca.cesarano@uni-bayreuth.de)
Luca Rizzi
Affiliation:
IBS Center for Complex Geometry, 55 EXPO-ro, Yuseong-gu, 34126 Daejeon, South Korea (lucarizzi@ibs.re.kr)
Francesco Zucconi
Affiliation:
DMIF, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy (francesco.zucconi@uniud.it)
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Abstract

Let $\pi \colon \mathcal {X}\to B$ be a family whose general fibre $X_b$ is a $(d_1,\,\ldots,\,d_a)$-polarization on a general abelian variety, where $1\leq d_i\leq 2$, $i=1,\,\ldots,\,a$ and $a\geq 4$. We show that the fibres are in the same birational class if all the $(m,\,0)$-forms on $X_b$ are liftable to $(m,\,0)$-forms on $\mathcal {X}$, where $m=1$ and $m=a-1$. Actually, we show a general criteria to establish whether the fibres of certain families belong to the same birational class.

Keywords

extension class of a vector bundleholomorphic formsAlbanese varietyfamilies of varietiesinfinitesimal invariant

MSC classification

Primary: 14J10: Families, moduli, classification 14C34: Torelli problem 14D07: Variation of Hodge structures
Secondary: 14E99: None of the above, but in this section 14J40: $n$-folds ($n>4$)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 3 , August 2022 , pp. 587 - 617
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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