论文标题
$ p = w $的列举方法
An Enumerative Approach to $P=W$
论文作者
论文摘要
$ p = w $猜想确定了希格斯（Higgs）模量空间的共同体的不正当过滤，以及相应角色品种的重量过滤。在本文中，我们引入了解决这个问题的列举方法。我们的技术仅使用Higgs束模量空间上的Equivariant交叉数字的结构，以及有关Hitchin Map拓扑的几乎没有信息。在$ 2 $的情况下，从稳定束的模量的已知交叉数开始，我们在Higgs Moduli上得出了均衡的交叉数字，然后验证我们枚举的$ P = W $语句的最高变态部分，即使是重训术类。该计算中的关键是存在满足特定消失特性的离散热方程的多项式溶液。对于奇数类，我们得出了枚举$ p = w $的确定标准。
The $P = W$ conjecture identifies the perverse filtration of the Hitchin system on the cohomology of the moduli space of Higgs bundles with the weight filtration of the corresponding character variety. In this paper, we introduce an enumerative approach to to this problem; our technique only uses the structure of the equivariant intersection numbers on the moduli space of Higgs bundles, and little information about the topology of the Hitchin map. In the rank $2$ case, starting from the known intersection numbers of the moduli of stable bundles, we derive the equivariant intersection numbers on the Higgs moduli, and then verify the top perversity part of our enumerative $P = W$ statement for even tautological classes. A key in this calculation is the existence of polynomial solutions to the Discrete Heat Equation satisfying particular vanishing properties. For odd classes, we derive a determinantal criterion for the enumerative $P = W$.
