The dual polarization of an abelian variety
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. \(\widehat H_1={1\over{(g-1)!}}\widehat H_2= {1\over d}\widehat H_3=\widehat H_4\). In this sense, they propose the name of dual polarization of \(H\) for the polarization \(\widehat H:=\widehat H_1=\widehat H_4\) on \(\widehat X\) and they compute the type of \(\widehat H\) in terms of the type of \(H\).
scientific article; zbMATH DE number 1383746