arXiv 1905.045672019-05math.AGhep-thmath-phmath.MP

K理论Donaldson-Thomas理论与曲面上点的Hilbert格式

K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface

Noah Arbesfeld

备注：: 56 pages, 10 figures, exposition expanded and section on rank two vector bundles added
发表信息：: (abridged version) Algebraic Geometry 8 (5) (2021) 587--625
DOI：: 10.14231/AG-2021-018

摘要：

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。研究了复曲面Calabi-Yau三重曲面的K-理论Donaldson-Thomas分拆函数在某些称为斜率的单参数子群下的极限，并给出了两个极限重合的条件.然后，我们显式地计算极限的分拆函数组件下所谓的首选斜率，获得明确的组合表达式有关的改进拓扑顶点的伊克巴尔，Kosçaz和Vafa。将这些结果应用于具体的Calabi-Yau三重性，我们推导出了由$\mathbb{C}^2$上的点的Hilbert方案上的重言式丛建立的生成函数所满足的对偶性。然后利用这种对偶性研究了一般曲面上点的Hilbert格式上重言丛的外幂和对称幂的全纯欧拉示性数。

英文摘要：

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, Kosçaz 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.

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