Rosati involution explained

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

\hat{A}=Pic^0(A)

be the dual abelian variety, and for

a\inA

, let

T_a:A\toA

be the translation-by-

map,

T_a(x)=x+a

. Then each divisor

defines a map

\phi_D:A\to\hatA

via

\phi_D(a)=[T

	*D-D]

	a

. The map

\phi_D

is a polarisation if

is ample. The Rosati involution of

End(A) ⊗ Q

relative to the polarisation

\phi_D

sends a map

\psi\inEnd(A) ⊗ Q

to the map

	-1
\psi'=\phi
	D

\circ\hat\psi\circ\phi_D

, where

\hat\psi:\hatA\to\hatA

is the dual map induced by the action of

\psi^*

Pic(A)

Let

NS(A)

denote the Néron–Severi group of

. The polarisation

\phi_D

also induces an inclusion

\Phi:NS(A) ⊗ Q\toEnd(A) ⊗ Q

via

\Phi_E=\phi

	-1

	D

\circ\phi_E

. The image of

\Phi

is equal to

\{\psi\inEnd(A) ⊗ Q:\psi'=\psi\}

, i.e., the set of endomorphisms fixed by the Rosati involution. The operation

E\starF=

	12\Phi
	^-1

(\Phi_E\circ\Phi_F+\Phi_F\circ\Phi_E)

then gives

NS(A) ⊗ Q

the structure of a formally real Jordan algebra