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.

Let $$A$$ be an abelian variety, let $$\hat{A} = \mathrm{Pic}^0(A)$$ be the dual abelian variety, and for $$a\in A$$, let $$T_a:A\to A$$ be the translation-by-$$a$$ map, $$T_a(x)=x+a$$. Then each divisor $$D$$ on $$A$$ defines a map $$\phi_D:A\to\hat A$$ via $$\phi_D(a)=[T_a^*D-D]$$. The map $$\phi_D$$ is a polarisation if $$D$$ is ample. The Rosati involution of $$\mathrm{End}(A)\otimes\mathbb{Q}$$ relative to the polarisation $$\phi_D$$ sends a map $$\psi\in\mathrm{End}(A)\otimes\mathbb{Q}$$ to the map $$\psi'=\phi_D^{-1}\circ\hat\psi\circ\phi_D$$, where $$\hat\psi:\hat A\to\hat A$$ is the dual map induced by the action of $$\psi^*$$ on $$\mathrm{Pic}(A)$$.

Let $$\mathrm{NS}(A)$$ denote the Néron–Severi group of $$A$$. The polarisation $$\phi_D$$ also induces an inclusion $$\Phi:\mathrm{NS}(A)\otimes\mathbb{Q}\to\mathrm{End}(A)\otimes\mathbb{Q}$$ via $$\Phi_E=\phi_D^{-1}\circ\phi_E$$. The image of $$\Phi$$ is equal to $$\{\psi\in\mathrm{End}(A)\otimes\mathbb{Q}:\psi'=\psi\}$$, i.e., the set of endomorphisms fixed by the Rosati involution. The operation $$E\star F=\frac12\Phi^{-1}(\Phi_E\circ\Phi_F+\Phi_F\circ\Phi_E)$$ then gives $$\mathrm{NS}(A)\otimes\mathbb{Q}$$ the structure of a formally real Jordan algebra.