论文标题
通用类型表面模量空间的虚拟基本类
The virtual fundamental class for the moduli space of surfaces of general type
论文作者
论文摘要
我们建议对索引模量堆栈的阻塞理论的构建，涵盖了一般类型的半神经典型表面。根据索引，涵盖了半循环表面的Deligne-Mumford堆栈，我们定义了覆盖Deligne-Mumford Stack的$ \ lci $。覆盖Deligne-Mumford堆栈的$ \ lci $仅具有本地完整的交叉点奇异点。 我们构建了$ \ lci $覆盖的模量堆栈，以便它允许适当的地图与通用类型表面的Moduli堆栈。然后，我们在LCI覆盖物的模量堆栈上构建了一个完美的阻塞理论，并在Moduli堆栈的Chow组上构建了一个虚拟基本类。因此，我们的建筑证明了西蒙·唐纳森爵士的猜想是在KSBA Moduli空间上存在虚拟基本班。重言式不变性是通过在虚拟基本阶层上的CM线捆绑组的第一类cm级捆绑的功能的整合来定义的。这可以作为对稳定曲线的稳定曲线的重言式类别$ \ overline {m} _g $在稳定表面的模量空间上的集成来定义的重言式不变术的概括。
We suggest a construction of obstruction theory on the moduli stack of index one covers over semi-log-canonical surfaces of general type. Based on the index one covering Deligne-Mumford stack of a semi-log-canonical surface, we define the $\lci$ covering Deligne-Mumford stack. The $\lci$ covering Deligne-Mumford stack only has locally complete intersection singularities. We construct the moduli stack of $\lci$ covers such that it admits a proper map to the moduli stack of surfaces of general type. We then construct a perfect obstruction theory on the moduli stack of lci covers and a virtual fundamental class on the Chow group of the moduli stack. Thus, our construction proves a conjecture of Sir Simon Donaldson for the existence of virtual fundamental class on the KSBA moduli spaces. A tautological invariant is defined by the integration of the power of the first Chern class for the CM line bundle of the moduli stack over the virtual fundamental class. This can be taken as a generalization of the tautological invariants defined by the integration of tautological classes over the moduli space $\overline{M}_g$ of stable curves to the moduli space of stable surfaces.
