The dual polarization of an abelian variety
type: Item
instance of: scholarly article
title: The dual polarization of an abelian variety
zbMATH Open document ID: 0979.14024
published in: Archiv der Mathematik
publication date: 2001-03-29T00:00:00Z
review text: Let \(X\) denote a polarized abelian variety of dimension \(g\) over the complex field \(\mathbb C\) and let \(L\) be an ample line bundle such that \(H=c_1(L)\in H^2(X,\mathbb Z)\) is the given polarization on \(X\), having degree \(d=h^0(L)\). We denote by \(\varphi_L:X\to\widehat X\) the isogeny induced by \(L\). The authors compare four different ways to associate to \(H\) a polarization \(\widehat H\) of the dual abelian variety \(\widehat X\) and they prove that these polarizations coincide up to a constant factor. Let us recall briefly the different constructions: (1) There exists a unique isogeny \(\psi:\widehat X\to X\), such that \(\psi\circ\varphi_L=d\cdot \text{id}_X\), and there is a unique polarization \(\widehat H_1=c_1(\widehat L_1)\) on \(\widehat X\), such that \(\psi=\varphi_{\widehat L_1}\). (2) Starting with the maps \(\mathbb L:H^r(X,\mathbb Z)\to H^{r+2}(X,\mathbb Z)\), defined by \(\varphi\mapsto c_1(L)\wedge\varphi\), one can consider the composed map \( H^2(X,\mathbb Z) @>\mathbb L^{g-2}>> H^{2g-2}(X,\mathbb Z) @>P^{-1}>>H_2(X,\mathbb Z) @>\delta_2>> H^2(\widehat X,\mathbb Z), \) where \(P\) denotes the Poincaré duality and \(\delta_2\) the canonical duality. The image of \(H\) under this map is the polarization of \(\widehat X\) denoted by \(\widehat H_2\). (3) Choosing a divisor \(D\) in the linear system \(|L|\), one gets a polarization \(\widehat H_3\), associated with the divisor \(\varphi_{L*}(D)\) and this polarization is independent of the choice of \(D\). (4) Denote by \(\mathcal P\) a Poincaré bundle on \(X\times\widehat X\) and take the Fourier-Mukai transform \({\mathcal F}(L)=p_{2*}({\mathcal P}\otimes p_1^{*}(L))\) on \(\widehat X\). This is a vector bundle of rank \(d\) and the line bundle \(\widehat L_4=\det({\mathcal F}(L))^{-1}\) defines a polarization \(\widehat H_4=c_1(\widehat L_4)\) on \(\widehat X\). The authors compute explicitly these different polarizations on \(\widehat X\) and they prove the following: Theorem. \…
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14K02
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 1383746
zbMATH Keywords: polarized abelian variety
zbMATH Keywords: isogeny
zbMATH Keywords: polarization
zbMATH Keywords: dual abelian variety
zbMATH Keywords: dual polarization
author: Christina Birkenhake
author: Herbert Lange
reviewed by: Maurizio Candilera
MaRDI profile type: Publication
full work available at URL: s000130050412
OpenAlex ID: W2094865706
Wikidata QID: Q102114785
DOI: 10.1007/S000130050412
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description: scientific article; zbMATH DE number 1383746