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 be an abelian variety, let
be the dual abelian variety, and for
, let
be the translation-by-
map,
. Then each divisor
on
defines a map
via
. The map
is a polarization, i.e., has finite kernel, if and only if
is ample. The Rosati involution of
relative to the polarization
sends a map
to the map
, where
is the dual map induced by the action of
on
.
Let denote the Néron–Severi group of
. The polarization
also induces an inclusion
via
. The image of
is equal to
, i.e., the set of endomorphisms fixed by the Rosati involution. The operation
then gives
the structure of a formally real Jordan algebra.
References
- Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290
- Rosati, 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