POLARIZED REAL TORI

Abstract

For a fixed positive integer g, we let <TEX>$\mathcal{P}_g=\{Y{\in}\mathbb{R}^{(g,g)}{\mid}Y=^tY</TEX><TEX>&</TEX><TEX>gt;0\}$</TEX> be the open convex cone in the Euclidean space <TEX>$\mathbb{R}^{g(g+1)/2}$</TEX>. Then the general linear group GL(g, <TEX>$\mathbb{R}$</TEX>) acts naturally on <TEX>$\mathcal{P}_g$</TEX> by <TEX>$A{\star}Y=AY^tA(A{\in}GL(g,\mathbb{R}),\;Y{\in}\mathcal{P}_g)$</TEX>. We introduce a notion of polarized real tori. We show that the open cone <TEX>$\mathcal{P}_g$</TEX> parametrizes principally polarized real tori of dimension g and that the Minkowski modular space 𝔗g = <TEX>$GL(g,\mathbb{Z}){\backslash}\mathcal{P}_g$</TEX> 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.