Tensor products of ample line bundles on abelian varieties
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 \((1,1,d)\).
type of a line bundle on an abelian variety
family of abelian threefolds admitting two principal polarizations
scientific article; zbMATH DE number 701149
scientific article