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  be an abelian variety, let  be the dual abelian variety, and for , let  be the translation-by- map, . Then each divisor  on  defines a map  via .  The map  is a polarisation if  is ample.  The Rosati involution of  relative to the polarisation   sends a map  to the map , where  is the dual map induced by the action of  on .

Let  denote the Néron–Severi group of . The polarisation  also induces an inclusion  via . The image of  is equal to , i.e., the set of endomorphisms fixed by the Rosati involution.  The operation  then gives  the structure of a formally real Jordan algebra.

References

Algebraic geometry
Ring theory