Rosati involution

Template:Short description 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 $\hat{A}={\mathrm{P}\mathrm{i}\mathrm{c}}^0(A)$ 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 ${\varphi }_D:A\to \hat{A}$ via ${\varphi }_D(a)=[T_a^{\ast }D-D]$. The map ${\varphi }_D$ is a polarisation if $D$ is ample. The Rosati involution of $\mathrm{E}\mathrm{n}\mathrm{d}(A)\otimes \mathbb{Q}$ relative to the polarisation ${\varphi }_D$ sends a map $\psi \in \mathrm{E}\mathrm{n}\mathrm{d}(A)\otimes \mathbb{Q}$ to the map ${\psi }^{\prime }={\varphi }_D^{-1}\circ \hat{\psi }\circ {\varphi }_D$, where $\hat{\psi }:\hat{A}\to \hat{A}$ is the dual map induced by the action of ${\psi }^{\ast }$ on $\mathrm{P}\mathrm{i}\mathrm{c}(A)$.

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

References

• Template:Citation
• Template:Citation