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 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 $E\star F=\frac12\Phi^{-1}(\Phi_E\circ\Phi_F+\Phi_F\circ\Phi_E)$ 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, MR 0282985, OCLC 138290
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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