Resolving mixed Hodge modules on configuration spaces
Resolving mixed Hodge modules on configuration spaces
复制标题
解决配置空间上的混合 Hodge 模块
DOI：
10.1215/s0012-7094-99-09605-9
复制
发表时间：
1996
影响因子：
2.5
通讯作者：
E. Getzler
中科院分区：
文献类型：
--
作者：
E. Getzler
本文给出了复数上方案X上的混合Hodge模E，以及拟射影态射f:X->S，构造了限制在f的第n个位形空间中的E的第n个外张量幂的自然分辨。这种构造使人联想到超平面排列理论中的技术，并依赖于Arnold对复线位形空间上同调的计算。这个分辨率是s_n等变的。将其应用于模曲线Y(N)上的阶数为N>=3的完备水平结构的通用椭圆曲线，得到模空间M_{1， N}的s_n -等变Serre多项式（混合Hodge结构类Grothendieck群中H^*_c（V，Q）的欧拉特征）的一个公式。在本文的续文中，将此应用于紧化\bar{M}_{1，n}的s_n -等变Hodge多项式的计算。
Given a mixed Hodge module E on a scheme X over the complex numbers, and a quasi-projective morphism f:X->S, we construct in this paper a natural resolution of the nth exterior tensor power of E restricted to the nth configuration space of f. The construction is reminiscent of techniques from the theory of hyperplane arrangements, and relies on Arnold's calculation of the cohomology of the configuration space of the complex line. This resolution is S_n-equivariant. We apply it to the universal elliptic curve with complete level structure of level N>=3 over the modular curve Y(N), obtaining a formula for the S_n-equivariant Serre polynomial (Euler characteristic of H^*_c(V,Q) in the Grothendieck group of the category of mixed Hodge structures) of the moduli space M_{1,n}. In a sequel to this paper, this is applied in the calculation of the S_n-equivariant Hodge polynomial of the compactication \bar{M}_{1,n}.