Journal of the
Korean Mathematical Society JKMS

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 $A\star Y=\,AY\,{}^t\!A$ ($A\in GL(g,{\mathbb R}),\ Y\in {\mathcal P}_g$). 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 modular space ${\mathfrak T}_g=\,GL(g,{\mathbb Z})\backslash {\mathcal P}_g$ 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.

Keywords: polarized real tori, line bundles over a real torus, semi-abelian varieties, semi-tori

MSC numbers: Primary 14K10