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 polarisation.

Let $A$ be an abelian variety, let $\hat = \mathrm^0(A)$ 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)= _a^*D-D/math>. The map $\phi_D$ is a polarisation if $D$ is ample. The Rosati involution of $\mathrm(A)\otimes\mathbb$ relative to the polarisation $\phi_D$ sends a map $\psi\in\mathrm(A)\otimes\mathbb$ to the map $\psi'=\phi_D^\circ\hat\psi\circ\phi_D$, where $\hat\psi:\hat A\to\hat A$ is the dual map induced by the action of $\psi^*$ on $\mathrm(A)$.

Let $\mathrm(A)$ denote the Néron–Severi group of $A$. The polarisation $\phi_D$ also induces an inclusion $\Phi:\mathrm(A)\otimes\mathbb\to\mathrm(A)\otimes\mathbb$ via $\Phi_E=\phi_D^\circ\phi_E$. The image of $\Phi$ is equal to $\$, i.e., the set of endomorphisms fixed by the Rosati involution. The operation $E\star F=\frac12\Phi^(\Phi_E\circ\Phi_F+\Phi_F\circ\Phi_E)$ then gives $\mathrm(A)\otimes\mathbb$ the structure of a formally real Jordan algebra.

References

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*{{Citation , last1=Rosati , first1=Carlo , title=Sulle corrispondenze algebriche fra i punti di due curve algebriche. , language=Italian , doi=10.1007/BF02419717 , year=1918 , journal=Annali di Matematica Pura ed Applicata , volume=3 , issue=28 , pages=35–60, s2cid=121620469 , url=https://zenodo.org/record/2226998

Algebraic geometry
Ring theory