POLARIZED REAL TORI
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For a fixed positive integer g, we let Pg = {Y ∈ R(g,g) | Y = tY > 0 } be the open convex cone in the Euclidean space Rg(g+1)/2. Then the general linear group GL(g, R) acts naturally on Pg by A⋆Y = AY tA (A ∈ GL(g, R), Y ∈ Pg). We introduce a notion of polarized real tori. We show that the open cone Pg parametrizes principally polarized real tori of dimension g and that the Minkowski modular space Tg = GL(g, Z)＼Pg 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.
Abstract
For a fixed positive integer g, we let Pg = {Y ∈ R(g,g) | Y = tY > 0 } be the open convex cone in the Euclidean space Rg(g+1)/2. Then the general linear group GL(g, R) acts naturally on Pg by A⋆Y = AY tA (A ∈ GL(g, R), Y ∈ Pg). We introduce a notion of polarized real tori. We show that the open cone Pg parametrizes principally polarized real tori of dimension g and that the Minkowski modular space Tg = GL(g, Z)＼Pg 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.
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