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 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 polarisation if is ample. The Rosati involution of relative to the polarisation 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 polarisation 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, vol. 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, S2CID 121620469
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