论文标题
虚拟$χ_ { - y} $ - 表面上引号方案的属
Virtual $χ_{-y}$-genera of Quot schemes on surfaces
论文作者
论文摘要
本文研究了Grothendieck在表面上的引号方案的虚拟$χ_ { - y} $ - 从而完善了Oprea-Pandharipande对虚拟Euler特征的计算。我们首先证明了一个结构性结果，表达了均值的虚拟虚拟$χ_ { - y} $ - 用seiberg-witten的不变性来普遍用引号的属。该公式更简单，对于纸张中定义的seiberg-witten长度$ n $的曲线类别。通过应用方式，我们在以下情况下给出了完整的答案：（i）零曲线类别的任意表面，（ii）纤维类别合理倍数的相对最小的椭圆形表面，（iii）一般类型的最小表面，$ p_g> $ p_g> 0 $ $ p_g> 0 $ $ p_g> 0 $。此外，对于Seiberg-witten Length $ n $的曲线类别，还获得了一个爆炸公式。由于这些计算，我们证明了虚拟$χ_ { - y} $ - 属的生成系列属于所有表面的有理函数，$ p_g> 0 $，解决了oprea-pandharipande的猜想。此外，我们研究了$ k3 $表面的$χ_ { - y} $ - 属，并与Kawai-yoshioka公式连接，$ k3 $表面和原始曲线类。
This paper studies the virtual $χ_{-y}$-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 $χ_{-y}$-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 blow up 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 $χ_{-y}$-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 $χ_{-y}$-genera for $K3$ surfaces and primitive curve classes with connections to the Kawai-Yoshioka formula.
