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 A be an abelian variety, let A ^ = P i c 0 ( A ) be the dual abelian variety, and for a ∈ A , let T a : A → A be the translation-by- a map, T a ( x ) = x + a . Then each divisor D on A defines a map ϕ D : A → A ^ via ϕ D ( a ) = [ T a ∗ D − D ] . The map ϕ D is a polarization, i.e., has finite kernel, if and only if D is ample. The Rosati involution of E n d ( A ) ⊗ Q relative to the polarization ϕ D sends a map ψ ∈ E n d ( A ) ⊗ Q to the map ψ ′ = ϕ D − 1 ∘ ψ ^ ∘ ϕ D , where ψ ^ : A ^ → A ^ is the dual map induced by the action of ψ ∗ on P i c ( A ) .
Let N S ( A ) denote the Néron–Severi group of A . The polarization ϕ D also induces an inclusion Φ : N S ( A ) ⊗ Q → E n d ( A ) ⊗ Q via Φ E = ϕ D − 1 ∘ ϕ E . The image of Φ is equal to { ψ ∈ E n d ( A ) ⊗ Q : ψ ′ = ψ } , i.e., the set of endomorphisms fixed by the Rosati involution. The operation E ⋆ F = 1 2 Φ − 1 ( Φ E ∘ Φ F + Φ F ∘ Φ E ) then gives N S ( A ) ⊗ Q the structure of a formally real Jordan algebra.
