Rosati involution

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

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 polarization, i.e., has finite kernel, if and only if $D$ is ample. The Rosati involution of ${\mathrm {End}}(A)\otimes {\mathbb {Q}}$ relative to the polarization $\phi _{D}$ 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 polarization $\phi _{D}$ 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 $E\star F={\frac 12}\Phi ^{{-1}}(\Phi _{E}\circ \Phi _{F}+\Phi _{F}\circ \Phi _{E})$ then gives ${\mathrm {NS}}(A)\otimes {\mathbb {Q}}$ the structure of a formally real Jordan algebra.

References

Mumford, David (2008) [1970], 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, MR 0282985
Rosati, Carlo (1918), "Sulle corrispondenze algebriche fra i punti di due curve algebriche.", Annali di Matematica Pura ed Applicata (in Italian) 3 (28): 35–60, doi:10.1007/BF02419717

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