Rosati involution

In mathematics, a Rosati involution, named after Carlo Rosati, is an involution of the rational endomorphism ring of an abelian variety induced by a polarisation.

Let $A$ be an abelian variety, let ${\hat {A}}=\mathrm {Pic} ^{0}(A)$ be the dual abelian variety, and for $a\in A$ , let $T_{a}:A\to A$ be the translation-by- $a$ map, $T_{a}(x)=x+a$ . Then each divisor $D$ on $A$ defines a map $\phi _{D}:A\to {\hat {A}}$ via $\phi _{D}(a)=[T_{a}^{*}D-D]$ . The map $\phi _{D}$ is a polarisation if $D$ is ample. The Rosati involution of $\mathrm {End} (A)\otimes \mathbb {Q}$ relative to the polarisation $\phi _{D}$ sends a map $\psi \in \mathrm {End} (A)\otimes \mathbb {Q}$ to the map $\psi '=\phi _{D}^{-1}\circ {\hat {\psi }}\circ \phi _{D}$ , where ${\hat {\psi }}:{\hat {A}}\to {\hat {A}}$ is the dual map induced by the action of $\psi ^{*}$ on $\mathrm {Pic} (A)$ .

Let $\mathrm {NS} (A)$ denote the Néron–Severi group of $A$ . The polarisation $\phi _{D}$ also induces an inclusion $\Phi :\mathrm {NS} (A)\otimes \mathbb {Q} \to \mathrm {End} (A)\otimes \mathbb {Q}$ via $\Phi _{E}=\phi _{D}^{-1}\circ \phi _{E}$ . The image of $\Phi$ is equal to $\{\psi \in \mathrm {End} (A)\otimes \mathbb {Q} :\psi '=\psi \}$ , i.e., the set of endomorphisms fixed by the Rosati involution. The operation $E\star F={\frac {1}{2}}\Phi ^{-1}(\Phi _{E}\circ \Phi _{F}+\Phi _{F}\circ \Phi _{E})$ then gives $\mathrm {NS} (A)\otimes \mathbb {Q}$ the structure of a formally real Jordan algebra.