Moduli spaces for real algebraic curves and real abelian varieties. (English) Zbl 0645.14012

In this paper we study moduli problems of real algebraic geometry in two ways. We first show that the space of real isomorphism classes of smooth real algebraic curves of a given topological type is a connected real analytic space. Using the Torelli mapping we then identify this moduli space with a subset of the moduli space of principally polarized real abelian varieties. This subset corresponds to real abelian varieties with a fixed number of connected components in the real part and a fixed real polarization (orthosymmetric or diasymmetric). We finally show that such abelian varieties form a semialgebraic set. This result allows us to equip also the moduli space of smooth real algebraic curves (of a given topological type) with a semialgebraic structure.
Reviewer: M.Seppälä


14Pxx Real algebraic and real-analytic geometry
14H10 Families, moduli of curves (algebraic)
14K10 Algebraic moduli of abelian varieties, classification
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