AI 中文总结
本文为广义Kummer型IHS簇的退化构造了分离对数紧化，通过热带化映射和热带广义Kummer构造，利用对数Hilbert概形推广了传统构造，并描述了边界纤维及多面体结构。
AI 中文摘要
我们构造了广义Kummer型极化不可约全纯辛（IHS）簇模空间中一个除子的分离对数紧化，该构造适用于维数至少为四的情形。此紧化带有一个对数IHS簇的普遍族，其热带化则带有一个热带广义Kummer簇的普遍族。该紧化基于一个更一般的构造，该构造为任意维数的广义Kummer簇在任意维数基对数概形上的II型和III型退化提供了显式、射影且对数光滑的模型。在任意对数秩的基上，这些退化配备了一个热带化映射，映射到一族带奇点的积分仿射流形，在II型情形下还带有边界。这一热带族通过热带广义Kummer构造而构建。我们的方法利用对数Hilbert概形，将传统的广义Kummer构造推广到对数阿贝尔簇的族。我们描述了退化的纤维，从而描述了紧化边界中的每一点。适当的模型对应于相关热带广义Kummer簇上的多面体结构，所得多面体复形作为对偶相交复形；每个不可约分量通过对半阿贝尔簇的对数群紧化之积应用广义Kummer构造而得到。
英文摘要
We construct a separated logarithmic compactification of a divisor in the moduli of polarised irreducible holomorphic symplectic (IHS) varieties of generalised Kummer type in dimension at least four. This compactification carries a universal family of logarithmic IHS varieties, and its tropicalisation carries a universal family of tropical generalised Kummer varieties. The compactification is built from a more general construction of explicit, projective, and logarithmically smooth models for type~II and type~III degenerations of generalised Kummer varieties of arbitrary dimension and over base logarithmic schemes of any dimension. Over a base of any logarithmic rank, these degenerations come with a tropicalisation map to a family of integral affine manifolds with singularities, and with boundary in the type II case. This tropical family is constructed through a tropical generalised Kummer construction. Our approach uses logarithmic Hilbert schemes to extend the conventional generalised Kummer construction to families of logarithmic abelian varieties. We describe fibres of our degenerations, and thus every point in the boundary of our compactification. Proper models correspond to polyhedral structures on the associated tropical generalised Kummer variety, with the resulting polyhedral complex serving as the dual intersection complex; each irreducible component is constructed by applying the generalised Kummer construction to a product of logarithmic group compactifications of semi-abelian varieties.
Comments75 pages, comments are welcome
