Squares of ample line bundles on abelian varieties
type: Item
instance of: scholarly article
title: Squares of ample line bundles on abelian varieties
zbMATH Open document ID: 0708.14029
author: Herbert Lange
author: M. S. Narasimhan
published in: Expositiones Mathematicae
publication date: 1989-01-01T00:00:00Z
review text: Let \(X\) be an abelian variety of dimension \(g\) defined over the complex field and let \(L\) be an ample line bundle. The rational mapping \(\phi =\Phi_{L^2}: X\to\mathbb{P}(H^0(L^2))\) associated with the complete linear system \(| L^2|\) is known to be a morphism. Let \((X,L) = (X_1,L_1)\times\cdots\times (X_s,L_s)\) be the decomposition into a product of irreducible polarized abelian varieties such that \((X_i,L_i)\) is principally polarized for \(1\le i\le r\) and not principally polarized for \(r+1\le i\le s\). Let \(K_i = X_i/(\pm 1)\) be the Kummer variety of \(X_i\) for \(1\le i\le r\) and \(p_i: X_i\to K_i\) the canonical projection. Let \(K=K_1\times\cdots\times K_r\times X_{r+1}\times\cdots\times X_ s\) and \(p=p_ 1\times...\times p_ r\times id_{X_{r+1}}\times\cdots\times id_{X_s}: X\to K\). Then it is shown that the morphism \(\phi\) decomposes as \(\psi \circ p: X\to K\to\mathbb{P}^N\) with an embedding \(\psi: K\hookrightarrow\mathbb{P}^N\). The result is a generalization of the results of \textit{S. Ramanan} [Proc. Lond. Math. Soc. (3) 51, 231--245 (1985; Zbl 0603.14013)], \textit{A. Ohbuchi} [Manuscr. Math. 57, 225--238 (1987; Zbl 0595.14013)], \textit{A. Andreotti} and \textit{A. L. Mayer} [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 21, 189--238 (1967; Zbl 0222.14024)] and \textit{D. Sasaki} [J. Reine Angew. Math. 345, 111--121 (1983; Zbl 0513.10027)].
Mathematics Subject Classification ID: 14K05
Mathematics Subject Classification ID: 14E25
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14F06
zbMATH DE Number: 4163067
zbMATH Keywords: principal polarization
zbMATH Keywords: complete linear system
zbMATH Keywords: irreducible polarized abelian varieties
zbMATH Keywords: Kummer variety
reviewed by: M. Miyanishi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4163067