Some Donaldson invariants of CP^2
Some Donaldson invariants of CP^2
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CP^2 的一些唐纳森不变量
DOI：
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
S. A. Strømme
中科院分区：
文献类型：
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作者：
G. Ellingsrud;J. L. Potier;S. A. Strømme
对于整数n≥2，设q4n−3为P = CP的4n−3次Donaldson多项式的系数。q4n−3在代数-几何上下文中的解释如下。设Mn表示P上秩为2、陈类c1 = 0、c2 = n的半稳定相干束的Gieseker-Maruyama模空间。对于这样的束F，利用grauert - m<s:1> lich定理推导出F对一般线L≤P的限制分裂为FL≤OL⊕OL，且这些例外线在对偶投影平面P上形成n次曲线J(F)。关联f7→J(F)是由称为Barth映射的代数变体的态射fn: Mn→Pn导出的。这里Pn = Pn (n+3)/2是参数化P中所有n次曲线的线性系统。设H∈Pic（Pn）为超平面类，设α = f * nH。Donaldson不变量的解释为：
For an integer n ≥ 2, let q4n−3 be the coefficient of the Donaldson polynomial of degree 4n − 3 of P = CP. An interpretation of q4n−3 in an algebro-geometric context is the following. Let Mn denote the Gieseker-Maruyama moduli space of semistable coherent sheaves on P with rank 2 and Chern classes c1 = 0 and c2 = n. For such a sheaf F , the Grauert-Mülich theorem implies that the restriction of F to a general line L ⊆ P splits as FL ≃ OL ⊕ OL, and that the exceptional lines form a curve J(F ) of degree n in the dual projective plane P . The association F 7→ J(F ) is induced from a morphism of algebraic varieties, called the Barth map, fn : Mn → Pn. Here Pn = P n(n+3)/2 is the linear system parameterizing all curves of degree n in P . Let H ∈ Pic(Pn) be the hyperplane class and let α = f ∗ nH . The interpretation of the Donaldson invariant is: