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
A
be an abelian variety, let
A
^
=
P
i
c
0
(
A
)
be the dual abelian variety, and for
a
∈
A
, let
T
a
:
A
→
A
be the translation-by-
a
map,
T
a
(
x
)
=
x
+
a
. Then each divisor
D
on
A
defines a map
ϕ
D
:
A
→
A
^
via
ϕ
D
(
a
)
=
[
T
a
∗
D
−
D
]
. The map
ϕ
D
is a polarization, i.e., has finite kernel, if and only if
D
is ample. The Rosati involution of
E
n
d
(
A
)
⊗
Q
relative to the polarization
ϕ
D
sends a map
ψ
∈
E
n
d
(
A
)
⊗
Q
to the map
ψ
′
=
ϕ
D
−
1
∘
ψ
^
∘
ϕ
D
, where
ψ
^
:
A
^
→
A
^
is the dual map induced by the action of
ψ
∗
on
P
i
c
(
A
)
.
Let
N
S
(
A
)
denote the Néron–Severi group of
A
. The polarization
ϕ
D
also induces an inclusion
Φ
:
N
S
(
A
)
⊗
Q
→
E
n
d
(
A
)
⊗
Q
via
Φ
E
=
ϕ
D
−
1
∘
ϕ
E
. The image of
Φ
is equal to
{
ψ
∈
E
n
d
(
A
)
⊗
Q
:
ψ
′
=
ψ
}
, 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
)
then gives
N
S
(
A
)
⊗
Q
the structure of a formally real Jordan algebra.
