Abstract

We introduce the notion of almost perfect obstruction theory on a Deligne–Mumford stack and show that stacks with almost perfect obstruction theories have virtual structure sheaves, which are deformation invariant. The main components in the construction are an induced embedding of the coarse moduli sheaf of the intrinsic normal cone into the associated obstruction sheaf stack and the construction of a $K$-theoretic Gysin map for sheaf stacks. We show that many stacks of interest admit almost perfect obstruction theories. As a result, we are able to define virtual structure sheaves and $K$-theoretic classical and generalized Donaldson–Thomas invariants of sheaves and complexes on Calabi–Yau three-folds.

中文翻译：

---

K-理论广义唐纳森-托马斯不变量

摘要

我们在 Deligne-Mumford 堆栈上引入了几乎完美的阻塞理论的概念，并表明具有几乎完美的阻塞理论的堆栈具有虚拟结构滑轮，它们是变形不变的。构造中的主要组成部分是将固有法向锥的粗模层诱导嵌入相关的障碍层堆中，以及构建层堆的$K$-理论 Gysin 映射。我们表明，许多感兴趣的堆栈都承认几乎完美的阻塞理论。因此，我们能够在 Calabi-Yau 三重上定义滑轮和复合体的虚拟结构滑轮和 $K$-理论经典和广义 Donaldson-Thomas 不变量。