Antiholomorphic Involutions of Analytic Families of Abelian Varieties

Abstract

Abstract.In this paper, we investigate antiholomorphic involutions of Kuga-Satake analytic families of polarized abelian varieties V. A complete set of invariants of the Aut(K)-conjugacy classes of antiholomorphic involutions of V is obtained. These invariants are expressed as cohomological invariants of the arithmetic data defining V. In the last section, the fibre varieties of Kuga-Satake type belonging to totally indefinite quaternion division algebras over totally real fields are investigated in more detail, and the cohomological invariants are related to results of Steve Kudla. The group of holomorphic sections of V is computed for this case. It is also shown that in general the fibre structure of V is intrinsic.

Real structures on ordinary Abelian varieties

“…Recall that a real structure on a complex Abelian variety A is an anti-holomorphic involution of A. It has been observed [50,51,7,49,39,1,17,15] that principally polarized Abelian varieties (of dimension n) with real structure correspond to "real points" of the moduli space X = Sp 2n (Z)\h n of all principally polarized Abelian varieties, where h n is the Siegel upper halfspace. On this variety, complex conjugation is induced from the mapping on h n that is given by Z → Z = − Z which is in turn induced from the "standard involution" τ 0 .…”

“…A fixed point in X therefore comes from a point Z ∈ h n such that Z = γZ for some γ ∈ Sp 2n (Z). By the Comessatti Lemma ( [50,51,7,15]) this implies that Z = 1 2 S + iY where S ∈ M n×n (Z) is a symmetric integral matrix, which may be taken to consist of zeroes and ones, and Y ∈ C n = GL n (R)/O(n) is an element of the cone of positive definite symmetric real matrices. Let {S 1 , S 2 , • • • , S r } be a collection of representatives of symmetric integral matrices consisting of zeroes and ones, modulo GL n (Z)-equivalence 3 .…”

“…) is an eigenvector of γ with eigenvalue λ then (a) τ 0 (w) = ( −u v ) is an eigenvector of γ with eigenvalue q/λ (b) u is an eigenvector of A with eigenvalue 1 2 λ + q λ (c) v is an eigenvector of t A with eigenvalue 1 2 λ + q λ .…”

“…Complex conjugation acts on the variety X and it has been observed ( [50,51,7,49,39,1,17,15]) that its fixed points (that is, the set of "real" points in X) correspond to (principally polarized) Abelian varieties that admit a real structure. A given Abelian variety may admit many distinct real structures.…”

“…Exploring this simple definition is the main object of this paper. Morphisms of Deligne modules with real structures are required to commute with the involutions 1 . In §3.8 we incorporate level structures.…”