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 polarisation.

be an abelian variety, let

{\displaystyle {\hat {A}}=\mathrm {Pic} ^{0}(A)}

be the dual abelian variety, and for

be the translation-by-

Then each divisor

defines a map

{\displaystyle \phi _{D}(a)=[T_{a}^{*}D-D]}

is a polarisation if

is ample.

The Rosati involution of

{\displaystyle \mathrm {End} (A)\otimes \mathbb {Q} }

relative to the polarisation

sends a map

{\displaystyle \psi \in \mathrm {End} (A)\otimes \mathbb {Q} }

is the dual map induced by the action of

{\displaystyle \mathrm {Pic} (A)}

{\displaystyle \mathrm {NS} (A)}

denote the Néron–Severi group of

The polarisation

also induces an inclusion

{\displaystyle \Phi :\mathrm {NS} (A)\otimes \mathbb {Q} \to \mathrm {End} (A)\otimes \mathbb {Q} }

The image of

is equal to

{\displaystyle \{\psi \in \mathrm {End} (A)\otimes \mathbb {Q} :\psi '=\psi \}}

, i.e., the set of endomorphisms fixed by the Rosati involution.

The operation

{\displaystyle \mathrm {NS} (A)\otimes \mathbb {Q} }

the structure of a formally real Jordan algebra.