# Polarized Real Tori - Mathematics > Algebraic Geometry

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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 theEuclidean space ${\mathbb R}^{gg+1-2}$. Then the general linear group$GLg,{\mathbb R}$ acts naturally on ${\mathcal P} g$ by $A\star Y= AY {}^tA$$A\in GLg,{\mathbb R}, Y\in {\mathcal P} g$. We introduce a notion ofpolarized real tori. We show that the open cone ${\mathcal P} g$ parametrizesprincipally polarized real tori of dimension $g$ and that the Minkowski domain${\mathfrak R} g= GLg,{\mathbb Z}\backslash {\mathcal P} g$ may be regardedas a moduli space of principally polarized real tori of dimension $g$. We alsostudy smooth line bundles on a polarized real torus by relating them toholomorphic line bundles on its associated polarized real abelian variety.

Author: ** Jae-Hyun Yang**

Source: https://arxiv.org/