- References
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
- {{Citation | last1=Mumford | first1=David | author1-link=David Mumford | title=Abelian varieties | origyear=1970 | publisher=American Mathematical Society | location=Providence, R.I. | series=Tata Institute of Fundamental Research Studies in Mathematics | isbn=978-81-85931-86-9 | oclc=138290 |mr=0282985 | year=2008 | volume=5}}
- {{Citation | last1=Rosati | first1=Carlo | title=Sulle corrispondenze algebriche fra i punti di due curve algebriche. | language=Italian | doi=10.1007/BF02419717 | year=1918 | journal=Annali di Matematica Pura ed Applicata | volume=3 | issue=28 | pages=35–60}}
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