Rosati involution

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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.