辛格上的球面复形
Spherical complexes attached to symplectic lattices
摘要：
对于整辛群Sp（2 g，Z），我们联系了两个偏序集，并证明了它们具有Cohen-Macaulay性质.作为应用，我们证明了亏格g的Siegel空间中的标记可分解主极化阿贝尔簇的轨迹具有一簇（g-2）-球面的同伦型.这反过来又意味着，有理的同源模空间的（未标记）主要极化阿贝尔簇的属g模的可分解的g-2度或更低的消失。另一个应用是欧氏环上的辛群的同调的改进的稳定范围。但最初的动机来自于对Torelli类型群的同调的设想应用。我们的主要结果的证明依赖于一个精炼的偏序集神经定理，它可能有自己的利益。
英文摘要：
To the integral symplectic group Sp(2g,Z) we associate two posets of which we prove that they have the Cohen-Macaulay property. As an application we show that the locus of marked decomposable principally polarized abelian varieties in the Siegel space of genus g has the homotopy type of a bouquet of (g-2)-spheres. This, in turn, implies that the rational homology of moduli space of (unmarked) principal polarized abelian varieties of genus g modulo the decomposable ones vanishes in degree g-2 or lower. Another application is an improved stability range for the homology of the symplectic groups over Euclidean rings. But the original motivation comes from envisaged applications to the homology of groups of Torelli type. The proof of our main result rests on a refined nerve theorem for posets that may have an interest in its own right.
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