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.
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, S2CID 121620469