## Journal of Differential Geometry

### Anti-holomorphic multiplication and a real algebraic modular variety

#### Abstract

An anti-holomorphic multiplication by the integers $O_d$ of a quadratic imaginary number field, on a principally polarized complex abelian variety $A_{\mathbb{C}}$ is an action of $O_d$ on $A_{\mathbb{C}}$ such that the purely imaginary elements act in an anti-holomorphic manner. The coarse moduli space $X_{\mathbb{R}}$ of such $A$ (with appropriate level structure) is shown to consist of finitely many isomorphic connected components, each of which is an arithmetic quotient of the quaternionic Siegel space, that is, the symmetric space for the complex sym- plectic group. The moduli space $X_{\mathbb{R}}$ is also identified as the fixed point set of a certain anti-holomorphic involution $\tau$ on the complex points $X_{\mathbb{C}}$ of the Siegel moduli space of all principally polarized abelian varieties (with appropriate level structure). The Siegel moduli space $X_{\mathbb{C}}$ admits a certain rational structure for which the involution $\tau$ is rationally defined. So the space $X_{\mathbb{R}}$ admits the structure of a rationally defined, real algebraic variety.

#### Article information

Source
J. Differential Geom., Volume 65, Number 3 (2003), 513-560.

Dates
First available in Project Euclid: 11 June 2015

https://projecteuclid.org/euclid.jdg/1434052758

Digital Object Identifier
doi:10.4310/jdg/1434052758

Mathematical Reviews number (MathSciNet)
MR2064430

Zentralblatt MATH identifier
1142.11040

#### Citation

Goresky, Mark; Tai, Yung Sheng. Anti-holomorphic multiplication and a real algebraic modular variety. J. Differential Geom. 65 (2003), no. 3, 513--560. doi:10.4310/jdg/1434052758. https://projecteuclid.org/euclid.jdg/1434052758