Tensor products of ample line bundles on abelian varieties
type: Item
instance of: scholarly article
title: Tensor products of ample line bundles on abelian varieties
zbMATH Open document ID: 0836.14026
DOI: 10.1007/BF02567440
published in: Manuscripta Mathematica
publication date: 1995-01-24T00:00:00Z
full work available at URL: https://eudml.org/doc/155975
review text: The type of a line bundle \(L\) on an abelian variety \(X = \mathbb{C}^g/ \Lambda\) is defined as the type of the first Chern class of \(L\) considered as an alternating form on the lattice \(\Lambda\). If \(L'\) is a second line bundle on \(X\), it is not easy in general to determine the type of the tensor product \(L \otimes L'\) in terms of \(L\) and \(L'\). The present paper gives an answer to this problem for abelian varieties of dimension 2 and 3. It turns out that the type of \(L \otimes L'\) not only depends on the types of \(L\) and \(L'\), but also on the intersection number \(L \cdot L'\) and the endomorphism ring \(\text{End} (X)\) of \(X\). Explicit formulas for these types are given. In order to apply these formulas one has to make sure that abelian varieties with given endomorphism ring exist. This is done for an arbitrary order in any totally real number field. Applying this and formulas of \textit{V. Ennola} and \textit{R. Turunen} [Math. Comput. 44, 495-518 (1985; Zbl 0564.12006)] on units in totally real number fields many examples are given. In particular using a paper by \textit{C. Birkenhake}, \textit{H. Lange} and \textit{S. Ramanan} [Manuscr. Math. 81, No. 3-4, 299-310 (1993; Zbl 0807.14030)] it is shown that for any \(d \geq 13\) there is a three-dimensional family of abelian threefolds admitting two principal polarizations whose tensor product is very ample of type \((1,1,d)\).
reviewed by: Herbert Lange
Mathematics Subject Classification ID: 14K05
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 701149
zbMATH Keywords: type of a line bundle on an abelian variety
zbMATH Keywords: family of abelian threefolds admitting two principal polarizations
author: Christina Birkenhake
MaRDI profile type: Publication
cites work: Joacobian surfaces in P4 .
cites work: On Totally Real Cubic Fields
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description: scientific article; zbMATH DE number 701149