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

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