Integral cohomology of Hilbert schemes of points on surfaces
Integral cohomology of Hilbert schemes of points on surfaces
复制标题
曲面上点的希尔伯特格式的积分上同调
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
Zhenbo Qin
中科院分区：
文献类型：
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作者：
Wei;Zhenbo Qin
投影曲面X上n个点的Hilbert格式X [n]的值可以用Göttsche、Grojnowski和Nakajima（[1-3]）的结果给出。当X是射影平面时，Ellingsrud和Strømme[4]发现了整上同调环H * （X [n]； Z）的一组环生成器，其本质上来自X [n]上某些重言束的Chern类（见[5]的推广）。如果X是一个任意投影曲面，则在[6]中找到了有理上同环H * (X [n])的一组环生成子（参见[7-11]）。由于在[6]中使用了重言束的陈氏特征，结果是有理系数。最近，Markman[12-14]利用积分上同环H∗（X [n]； Z）研究了当X为K3曲面，Y为等价于X [n]的hyperkähler流形变形时H2（Y； Z）上的权-2 Hodge结构。在一般曲面X上，寻找整上同环H * （X [n]； Z）的可加基和环生成器是一个有趣的问题。主要思想是研究积分算子，即H * (X [n])上的线性算子，它将H * (X [n])中的积分类转换为积分类。
of the Hilbert scheme X [n] of n points on a projective surface X can be given using the results of Göttsche, Grojnowski and Nakajima ([1–3]). When X is a projective plane, Ellingsrud and Strømme [4] found a set of ring generators for the integral cohomology ring H∗(X [n]; Z), which essentially comes from the Chern classes of certain tautological bundles over X [n] (see [5] for a generalization). If X is an arbitrary projective surface, a set of ring generators for the rational cohomology ring H∗(X [n]) is found in [6] (see also [7–11]). The result is for rational coefficients since the Chern characters of the tautological bundles were used in [6]. Recently, Markman [12–14] used the integral cohomology ring H∗(X [n]; Z) to study the weight-2 Hodge structure on H2(Y ; Z) when X is a K3 surface and Y is a hyperkähler manifold deformation equivalent to X [n]. It is interesting to search for additive bases and ring generators of the integral cohomology ring H∗(X [n]; Z) for a general surface X. In [15], an effective method for finding integral additive bases was developed. The main idea there is to study integral operators, i.e., linear operators on H∗(X [n]) which send integral classes in H∗(X [n]) to integral ones.