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 polarization.

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 polarization, i.e., has finite kernel, if and only if D is ample. The Rosati involution of \mathrm{End}(A)\otimes\mathbb{Q} relative to the polarization \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 polarization \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.

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