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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 polarisation.

Let A {\displaystyle A} be an abelian variety, let A ^ = P i c 0 ( A ) {\displaystyle {\hat {A}}=\mathrm {Pic} ^{0}(A)} be the dual abelian variety, and for a ∈ A {\displaystyle a\in A} , let T a : A → A {\displaystyle T_{a}:A\to A} be the translation-by- a {\displaystyle a} map, T a ( x ) = x + a {\displaystyle T_{a}(x)=x+a} . Then each divisor D {\displaystyle D} on A {\displaystyle A} defines a map ϕ D : A → A ^ {\displaystyle \phi _{D}:A\to {\hat {A}}} via ϕ D ( a ) = [ T a ∗ D − D ] {\displaystyle \phi _{D}(a)=[T_{a}^{*}D-D]} . The map ϕ D {\displaystyle \phi _{D}} is a polarisation if D {\displaystyle D} is ample. The Rosati involution of E n d ( A ) ⊗ Q {\displaystyle \mathrm {End} (A)\otimes \mathbb {Q} } relative to the polarisation ϕ D {\displaystyle \phi _{D}} sends a map ψ ∈ E n d ( A ) ⊗ Q {\displaystyle \psi \in \mathrm {End} (A)\otimes \mathbb {Q} } to the map ψ ′ = ϕ D − 1 ∘ ψ ^ ∘ ϕ D {\displaystyle \psi '=\phi _{D}^{-1}\circ {\hat {\psi }}\circ \phi _{D}} , where ψ ^ : A ^ → A ^ {\displaystyle {\hat {\psi }}:{\hat {A}}\to {\hat {A}}} is the dual map induced by the action of ψ ∗ {\displaystyle \psi ^{*}} on P i c ( A ) {\displaystyle \mathrm {Pic} (A)} .

Let N S ( A ) {\displaystyle \mathrm {NS} (A)} denote the Néron–Severi group of A {\displaystyle A} . The polarisation ϕ D {\displaystyle \phi _{D}} also induces an inclusion Φ : N S ( A ) ⊗ Q → E n d ( A ) ⊗ Q {\displaystyle \Phi :\mathrm {NS} (A)\otimes \mathbb {Q} \to \mathrm {End} (A)\otimes \mathbb {Q} } via Φ E = ϕ D − 1 ∘ ϕ E {\displaystyle \Phi _{E}=\phi _{D}^{-1}\circ \phi _{E}} . The image of Φ {\displaystyle \Phi } is equal to { ψ ∈ E n d ( A ) ⊗ Q : ψ ′ = ψ } {\displaystyle \{\psi \in \mathrm {End} (A)\otimes \mathbb {Q} :\psi '=\psi \}} , 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 ) {\displaystyle E\star F={\frac {1}{2}}\Phi ^{-1}(\Phi _{E}\circ \Phi _{F}+\Phi _{F}\circ \Phi _{E})} then gives N S ( A ) ⊗ Q {\displaystyle \mathrm {NS} (A)\otimes \mathbb {Q} } the structure of a formally real Jordan algebra.

• citerefmumford2008Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290

• citerefrosati1918Rosati, 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