论文标题
真正的模量空间和非简单的真实阿伯利亚品种的密度
Real moduli spaces and density of non-simple real abelian varieties
论文作者
论文摘要
对于固定的$ k <g $和一个两极分化的尺寸$ g $ $ g $ of $ \ mathbb {r} $的家族，我们给出了这些亚伯利亚品种超过$ \ mathbb {r} $的参数空间中的密度标准，包含$ k $ k $ k $ - dimemensional abelian abelian abelian subvarian cubvarian $ mathbb。作为应用，我们证明了这种集合在极化的真实Abelian dimension $ g $的模量空间中的密度，并且在实际代数曲线和真实平面曲线的真实代数空间中，非繁琐的真实代数曲线的密度非绘制到真实的$ k $ j $ j $二维的Abelian品种。这扩展到了科伦坡和皮罗拉的真实设置结果，正如其论文中概述的“具有非简单雅各布人的曲线的某些密度结果”，$ \ textit {Math。 Ann。} $（1990）。然后，我们考虑$ \ mathbb {r} $上的代数堆栈的真实源，并将拓扑空间附加到它。对于真实的模量堆栈，这定义了一个真实的模量空间。我们表明，对于$ \ Mathcal {M} _g $和$ \ Mathcal {a} _g $，以这种方式出现的真实模量空间与Gross-Harris的Moduli空间一致（“真实代数曲线”，“真实代数曲线”，$ \ textit {Ann。Sci。Ecole。sepp。 （“用于真实代数曲线和真实Abelian品种的Moduli空间”，$ \ textit {Math。Z。} $（1989））。
For fixed $k<g$ and a family of polarized abelian varieties of dimension $g$ over $\mathbb{R}$, we give a criterion for the density in the parameter space of those abelian varieties over $\mathbb{R}$ containing a $k$-dimensional abelian subvariety over $\mathbb{R}$. As application, we prove density of such a set in the moduli space of polarized real abelian varieties of dimension $g$, and density of real algebraic curves mapping non-trivially to real $k$-dimensional abelian varieties in the moduli space of real algebraic curves as well as in the space of real plane curves. This extends to the real setting results by Colombo and Pirola as outlined in their paper "Some density results for curves with non-simple jacobians", $\textit{Math. Ann.}$ (1990). We then consider the real locus of an algebraic stack over $\mathbb{R}$, attaching a topological space to it. For a real moduli stack, this defines a real moduli space. We show that for $\mathcal{M}_g$ and $\mathcal{A}_g$, the real moduli spaces that arise in this way coincide with the moduli spaces of Gross-Harris ("Real algebraic curves", $\textit{Ann. Sci. Ecole. Norm. Sup.}$ (1981)) and Seppälä-Silhol ("Moduli Spaces for Real Algebraic Curves and Real Abelian Varieties", $\textit{Math. Z.}$ (1989)).
