PERVERSE SHEAVES AND THE COHOMOLOGY OF HILBERT SCHEMES OF SMOOTH ALGEBRAIC-SURFACES
PERVERSE SHEAVES AND THE COHOMOLOGY OF HILBERT SCHEMES OF SMOOTH ALGEBRAIC-SURFACES
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DOI：
10.1007/bf01445104
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发表时间：
1993-06-01
影响因子：
1.4
通讯作者：
SOERGEL, W
中科院分区：
文献类型：
--
作者：
GOTTSCHE, L;SOERGEL, W
对于任何代数簇（即有限型的分离概型）X/C，令X（C）表示闭点集及其解析拓扑。对任意光滑代数曲面S/C，n= 1，2，…可以将第n个希尔伯特方案S [n]参数化S的长度为n的封闭零维子方案（参见图1）。[Gr，F]）。已知S tn]是维数为2n的光滑的（参见图1）。定理2.4]）。我们想展示如何从曲面本身的奇异上同调H*（S（C），Q）计算Hilbert格式的奇异上同调H~（Q）。在第6节中，我们将以类似的方式计算高阶库默簇的奇异上同调。为了至少给出希尔伯特格式的结果，我们必须引入一些符号。设P（n）是n的分拆集。我们将cu E P（n）记为n-c~ 1 9 1+.+ c~。r和put] c~[=~ ai.设q（n）是S的n次对称幂。这是众所周知的（cf。[Mac]）如何从H~ Q）计算H~ Q），即作为对称群~ n的不变量置换H '（S（C），Q）的因子|合适的标志。对于c~ EP（n），我们将S（1 ~）：S（al）·..·6“（"”）。我们的主要结果的第一近似如下。
For any algebraic variety (ie a separated scheme of finite type) X/C let X (C) denote the set of closed points with its analytic topology. To any smooth algebraic surface S/C and n= 1, 2,... one may associate the n'th Hilbert scheme S [n] parametrizing closed zero-dimensional subschemes of length n of S (cf.[Gr, F]). It is known that S tn] is smooth of dimension 2n (cf. IF, Theorem 2.4]). We want to show how the singular cohomology H~ Q) of the Hilbert scheme can be computed from the singular cohomology H*(S (C), Q) of the surface itself. In Sect. 6 we will in a similar way compute the singular cohomology of higher order Kummer varieties. To give at least the result for Hilbert schemes, we have to introduce some notations. Let P (n) be the set of partitions of n. We write cu E P (n) as n---c~ 1 9 1+...+ c~. r and put] c~[=~ ai. Let~ q (n) be the n'th symmetric power of S. It is well known (cf.[Mac]) how to calculate H~ Q) from H~ Q), namely as the invariants of the symmetric group~ n permuting the factors of H'(S (C), Q)| with suitable signs. For c~ E P (n) we put S ('~): S (al)•...• 6'('~'). A first approximation of our main result is the following.