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 [math]\displaystyle{ A }[/math] be an abelian variety, let [math]\displaystyle{ \hat{A} = \mathrm{Pic}^0(A) }[/math] be the dual abelian variety, and for [math]\displaystyle{ a\in A }[/math], let [math]\displaystyle{ T_a:A\to A }[/math] be the translation-by-[math]\displaystyle{ a }[/math] map, [math]\displaystyle{ T_a(x)=x+a }[/math]. Then each divisor [math]\displaystyle{ D }[/math] on [math]\displaystyle{ A }[/math] defines a map [math]\displaystyle{ \phi_D:A\to\hat A }[/math] via [math]\displaystyle{ \phi_D(a)=[T_a^*D-D] }[/math]. The map [math]\displaystyle{ \phi_D }[/math] is a polarisation if [math]\displaystyle{ D }[/math] is ample. The Rosati involution of [math]\displaystyle{ \mathrm{End}(A)\otimes\mathbb{Q} }[/math] relative to the polarisation [math]\displaystyle{ \phi_D }[/math] sends a map [math]\displaystyle{ \psi\in\mathrm{End}(A)\otimes\mathbb{Q} }[/math] to the map [math]\displaystyle{ \psi'=\phi_D^{-1}\circ\hat\psi\circ\phi_D }[/math], where [math]\displaystyle{ \hat\psi:\hat A\to\hat A }[/math] is the dual map induced by the action of [math]\displaystyle{ \psi^* }[/math] on [math]\displaystyle{ \mathrm{Pic}(A) }[/math].
Let [math]\displaystyle{ \mathrm{NS}(A) }[/math] denote the Néron–Severi group of [math]\displaystyle{ A }[/math]. The polarisation [math]\displaystyle{ \phi_D }[/math] also induces an inclusion [math]\displaystyle{ \Phi:\mathrm{NS}(A)\otimes\mathbb{Q}\to\mathrm{End}(A)\otimes\mathbb{Q} }[/math] via [math]\displaystyle{ \Phi_E=\phi_D^{-1}\circ\phi_E }[/math]. The image of [math]\displaystyle{ \Phi }[/math] is equal to [math]\displaystyle{ \{\psi\in\mathrm{End}(A)\otimes\mathbb{Q}:\psi'=\psi\} }[/math], i.e., the set of endomorphisms fixed by the Rosati involution. The operation [math]\displaystyle{ E\star F=\frac12\Phi^{-1}(\Phi_E\circ\Phi_F+\Phi_F\circ\Phi_E) }[/math] then gives [math]\displaystyle{ \mathrm{NS}(A)\otimes\mathbb{Q} }[/math] the structure of a formally real Jordan algebra.
References
- Mumford, David (2008), Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, OCLC 138290
- Rosati, Carlo (1918), "Sulle corrispondenze algebriche fra i punti di due curve algebriche." (in Italian), Annali di Matematica Pura ed Applicata 3 (28): 35–60, doi:10.1007/BF02419717, https://zenodo.org/record/2226998
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