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 $\hat{A}={\mathrm{P}\mathrm{i}\mathrm{c}}^0(A)$ 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 ${\varphi }_D(a)=[T_a^{*}D-D]$. The map ${\varphi }_D$ is a polarisation if $D$ is ample. The Rosati involution of $\mathrm{E}\mathrm{n}\mathrm{d}(A)\otimes \mathbb{\mathbb{Q}}$ relative to the polarisation ${\varphi }_D$ sends a map $\psi \in \mathrm{E}\mathrm{n}\mathrm{d}(A)\otimes \mathbb{\mathbb{Q}}$ 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 ${\psi }^{*}$ 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 $\mathrm{\Phi }:\mathrm{N}\mathrm{S}(A)\otimes \mathbb{\mathbb{Q}}\to \mathrm{E}\mathrm{n}\mathrm{d}(A)\otimes \mathbb{\mathbb{Q}}$ via ${\mathrm{\Phi }}_E={\varphi }_D^{-1}\circ {\varphi }_E$. The image of $\mathrm{\Phi }$ is equal to $\{\psi \in \mathrm{E}\mathrm{n}\mathrm{d}(A)\otimes \mathbb{\mathbb{Q}}:{\psi }^{\prime }=\psi \}$, 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 $\mathrm{N}\mathrm{S}(A)\otimes \mathbb{\mathbb{Q}}$ the structure of a formally real Jordan algebra.

References

• Mumford, David (2008), Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, OCLC 138290 
• Rosati, Carlo (1918), "Sulle corrispondenze algebriche fra i punti di due curve algebriche." (in Italian), Annali di Matematica Pura ed Applicata 3 (28): 35–60, doi:10.1007/BF02419717, https://zenodo.org/record/2226998