Polarized Real Tori

Abstract

For a fixed positive integer g, we let Pg = Y ∈ R (g,g) Y = t Y > 0 be the open convex cone in the Euclidean space R g(g+1) 2 . Then the general linear group GL(g, R) acts naturally on Pg by A ⋆ Y = AY t A (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 domain Rg = 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.2000 Mathematics Subject Classification: Primary 14K10 Keywords and phrases: Real abelian varieties, moduli space of real principally polarized abelian varieties, polarized real tori, line bundles over a real torus, semi-abelian varieties, semi-tori.

Invariant Differential Operators on the Minkowski-Euclid Space

“…The arithmetic quotient Γ n,m \P n,m may be regarded as the universal family of principally polarized real tori of dimension mn (cf. [13]). I propose to name the space P n,m the Minkowski-Euclid space because H. Minkowski [8] found a fundamental domain for P n with respect to the arithmetic subgroup GL(n, Z) by means of the reduction theory.…”