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 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 polarization, i.e., has finite kernel, if and only if is ample. The Rosati involution of relative to the polarization 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 polarization 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.
