K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface
K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface
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DOI：
10.14231/ag-2021-018
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发表时间：
2019-05
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影响因子：
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通讯作者：
N. Arbesfeld
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文献类型：
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作者：
N. Arbesfeld
曲面上点的希尔伯特格式上重言层的特征类的积分经常出现在计数问题中。我们使用K理论的Donaldson-Thomas理论的某些复曲面卡-丘三倍，研究K理论的变种，这样的表达式。我们研究的K-理论Donaldson-Thomas配分函数的复曲面Calabi-Yau三倍在某些单参数子群称为斜坡下的极限，并制定了一个条件下，两个这样的限制重合。然后，我们显式地计算的限制下，所谓的首选斜率的配分函数的组件，获得明确的组合表达式有关的精炼拓扑顶点的伊克巴尔，Koscaz和Vafa。将这些结果应用于特定的卡-丘三重，我们推导出由$\mathbb{C}^2 $上的点的希尔伯特方案上的重言丛建立的生成函数所满足的对偶。然后，我们使用这种对偶研究全纯欧拉特征的外部和对称的权力重言式丛的希尔伯特计划的点在一般的表面上。
Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds to study K-theoretic variants of such expressions. We study limits of the K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold under certain one-parameter subgroups called slopes, and formulate a condition under which two such limits coincide. We then explicitly compute the limits of components of the partition function under so-called preferred slopes, obtaining explicit combinatorial expressions related to the refined topological vertex of Iqbal, Koscaz and Vafa. Applying these results to specific Calabi-Yau threefolds, we deduce dualities satisfied by a generating function built from tautological bundles on the Hilbert scheme of points on $\mathbb{C}^2$. We then use this duality to study holomorphic Euler characteristics of exterior and symmetric powers of tautological bundles on the Hilbert scheme of points on a general surface.