Polarized Real Tori

TL;DR

This paper introduces polarized real tori, showing their parametrization by a convex cone and relating their line bundles to those on associated real abelian varieties, thus establishing a moduli space framework.

Contribution

It defines polarized real tori, connects them to the convex cone ${ m f P}_g$, and constructs a moduli space using Minkowski domain, linking line bundles to real abelian varieties.

Findings

${ m f P}_g$ parametrizes principally polarized real tori.

The Minkowski domain ${ m f R}_g$ serves as a moduli space.

Line bundles on real tori relate to holomorphic line bundles on abelian varieties.

Abstract

For a fixed positive integer $g$, we let ${\mathcal P}_g = \big\{Y\in {\mathbb R}^{(g,g)} | Y= {}^tY>0 \big\}$ be the open convex cone in the Euclidean space ${\mathbb R}^{g(g+1)/2}$. Then the general linear group $GL(g,{\mathbb R})$ acts naturally on ${\mathcal P}_g$ by $A\star Y= AY {}^tA$ ($A\in GL(g,{\mathbb R}), Y\in {\mathcal P}_g$). We introduce a notion of polarized real tori. We show that the open cone ${\mathcal P}_g$ parametrizes principally polarized real tori of dimension $g$ and that the Minkowski domain ${\mathfrak R}_g= GL(g,{\mathbb Z})\backslash {\mathcal P}_g$ may be regarded as a moduli space of principally polarized real tori of dimension $g$. We also study smooth line bundles on a polarized real torus by relating them to holomorphic line bundles on its associated polarized real abelian variety.