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Group theoretic operation
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
A
{\displaystyle A}
be an abelian variety , let
A
^
=
P
i
c
0
(
A
)
{\displaystyle {\hat {A}}=\mathrm {Pic} ^{0}(A)}
be the dual abelian variety , and for
a
∈
A
{\displaystyle a\in A}
, let
T
a
:
A
→
A
{\displaystyle T_{a}:A\to A}
be the translation-by-
a
{\displaystyle a}
map,
T
a
(
x
)
=
x
+
a
{\displaystyle T_{a}(x)=x+a}
. Then each divisor
D
{\displaystyle D}
on
A
{\displaystyle A}
defines a map
ϕ
D
:
A
→
A
^
{\displaystyle \phi _{D}:A\to {\hat {A}}}
via
ϕ
D
(
a
)
=
[
T
a
∗
D
−
D
]
{\displaystyle \phi _{D}(a)=[T_{a}^{*}D-D]}
. The map
ϕ
D
{\displaystyle \phi _{D}}
is a polarisation if
D
{\displaystyle D}
is ample . The Rosati involution of
E
n
d
(
A
)
⊗
Q
{\displaystyle \mathrm {End} (A)\otimes \mathbb {Q} }
relative to the polarisation
ϕ
D
{\displaystyle \phi _{D}}
sends a map
ψ
∈
E
n
d
(
A
)
⊗
Q
{\displaystyle \psi \in \mathrm {End} (A)\otimes \mathbb {Q} }
to the map
ψ
′
=
ϕ
D
−
1
∘
ψ
^
∘
ϕ
D
{\displaystyle \psi '=\phi _{D}^{-1}\circ {\hat {\psi }}\circ \phi _{D}}
, where
ψ
^
:
A
^
→
A
^
{\displaystyle {\hat {\psi }}:{\hat {A}}\to {\hat {A}}}
is the dual map induced by the action of
ψ
∗
{\displaystyle \psi ^{*}}
on
P
i
c
(
A
)
{\displaystyle \mathrm {Pic} (A)}
.
Let
N
S
(
A
)
{\displaystyle \mathrm {NS} (A)}
denote the Néron–Severi group of
A
{\displaystyle A}
. The polarisation
ϕ
D
{\displaystyle \phi _{D}}
also induces an inclusion
Φ
:
N
S
(
A
)
⊗
Q
→
E
n
d
(
A
)
⊗
Q
{\displaystyle \Phi :\mathrm {NS} (A)\otimes \mathbb {Q} \to \mathrm {End} (A)\otimes \mathbb {Q} }
via
Φ
E
=
ϕ
D
−
1
∘
ϕ
E
{\displaystyle \Phi _{E}=\phi _{D}^{-1}\circ \phi _{E}}
. The image of
Φ
{\displaystyle \Phi }
is equal to
{
ψ
∈
E
n
d
(
A
)
⊗
Q
:
ψ
′
=
ψ
}
{\displaystyle \{\psi \in \mathrm {End} (A)\otimes \mathbb {Q} :\psi '=\psi \}}
, i.e., the set of endomorphisms fixed by the Rosati involution. The operation
E
⋆
F
=
1
2
Φ
−
1
(
Φ
E
∘
Φ
F
+
Φ
F
∘
Φ
E
)
{\displaystyle E\star F={\frac {1}{2}}\Phi ^{-1}(\Phi _{E}\circ \Phi _{F}+\Phi _{F}\circ \Phi _{E})}
then gives
N
S
(
A
)
⊗
Q
{\displaystyle \mathrm {NS} (A)\otimes \mathbb {Q} }
the structure of a formally real Jordan algebra .
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