This paper studies the virtual ${\chi }_{-y}$-genera of Grothendieck's Quot schemes on surfaces, thus refining the calculations of the virtual Euler characteristics by Oprea–Pandharipande. We first prove a structural result expressing the equivariant virtual ${\chi }_{-y}$-genera of Quot schemes universally in terms of the Seiberg–Witten invariants. The formula is simpler for curve classes of Seiberg–Witten length $N$, which are defined in the paper. By way of application, we give complete answers in the following cases:

• (i) arbitrary surfaces for the zero curve class;
• (ii) relatively minimal elliptic surfaces for rational multiples of the fiber class;
• (iii) minimal surfaces of general type with $p_g>0$ for any curve classes.

Furthermore, a blowup formula is obtained for curve classes of Seiberg–Witten length $N$. As a result of these calculations, we prove that the generating series of the virtual ${\chi }_{-y}$-genera are given by rational functions for all surfaces with $p_g>0$, addressing a conjecture of Oprea–Pandharipande. In addition, we study the reduced ${\chi }_{-y}$-genera for $K3$ surfaces and primitive curve classes with connections to the Kawai–Yoshioka formula.

中文翻译：

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表面上 Quot 方案的虚拟 χ−y 属

本文研究了虚拟 ${\chi }_{-是}$-格洛腾迪克在表面上的报价方案的属，从而改进了 Oprea-Pandharipande 对虚拟欧拉特征的计算。我们首先证明了表达等变虚拟的结构结果${\chi }_{-是}$- 根据 Seiberg-Witten 不变量，Quot 方案的种类普遍存在。对于 Seiberg-Witten 长度的曲线类，公式更简单$N$，在论文中定义。通过应用，我们在以下情况下给出完整的答案：

• (i)零曲线类的任意曲面；
• (ii)纤维类有理倍数的相对最小的椭圆表面；
• (iii)具有一般类型的最小表面${磷}_G>0$ 对于任何曲线类。

此外，获得了 Seiberg-Witten 长度曲线类的膨胀公式 $N$. 作为这些计算的结果，我们证明了虚拟的生成级数${\chi }_{-是}$-genera 由所有曲面的有理函数给出 ${磷}_G>0$，解决 Oprea-Pandharipande 的猜想。此外，我们研究了减少的${\chi }_{-是}$-属为 $钾3$ 与 Kawai-Yoshioka 公式相关的曲面和原始曲线类。