Rosati involution

{{Short description|Group theoretic operation}}

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 $\backslash hat\{A\}\; =\; \backslash mathrm\{Pic\}^0(A)$ be the dual abelian variety, and for $a\backslash in\; A$, let $T\_a:A\backslash to\; A$ be the translation-by-$a$ map, $T\_a(x)=x+a$. Then each divisor $D$ on $A$ defines a map $\backslash phi\_D:A\backslash to\backslash hat\; A$ via $\backslash phi\_D(a)=[T\_a^*D-D]$. The map $\backslash phi\_D$ is a polarisation if $D$ is ample. The Rosati involution of $\backslash mathrm\{End\}(A)\backslash otimes\backslash mathbb\{Q\}$ relative to the polarisation $\backslash phi\_D$ sends a map $\backslash psi\backslash in\backslash mathrm\{End\}(A)\backslash otimes\backslash mathbb\{Q\}$ to the map $\backslash psi\text{'}=\backslash phi\_D^\{-1\}\backslash circ\backslash hat\backslash psi\backslash circ\backslash phi\_D$, where $\backslash hat\backslash psi:\backslash hat\; A\backslash to\backslash hat\; A$ is the dual map induced by the action of $\backslash psi^*$ on $\backslash mathrm\{Pic\}(A)$.

Let $\backslash mathrm\{NS\}(A)$ denote the Néron–Severi group of $A$. The polarisation $\backslash phi\_D$ also induces an inclusion $\backslash Phi:\backslash mathrm\{NS\}(A)\backslash otimes\backslash mathbb\{Q\}\backslash to\backslash mathrm\{End\}(A)\backslash otimes\backslash mathbb\{Q\}$ via $\backslash Phi\_E=\backslash phi\_D^\{-1\}\backslash circ\backslash phi\_E$. The image of $\backslash Phi$ is equal to $\backslash \{\backslash psi\backslash in\backslash mathrm\{End\}(A)\backslash otimes\backslash mathbb\{Q\}:\backslash psi\text{'}=\backslash psi\backslash \}$, i.e., the set of endomorphisms fixed by the Rosati involution. The operation $E\backslash star\; F=\backslash frac12\backslash Phi^\{-1\}(\backslash Phi\_E\backslash circ\backslash Phi\_F+\backslash Phi\_F\backslash circ\backslash Phi\_E)$ then gives $\backslash mathrm\{NS\}(A)\backslash otimes\backslash mathbb\{Q\}$ the structure of a formally real Jordan algebra.