Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and K-theoretic invariants for many moduli stacks of interest, including K-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the K-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory. To this end, we further investigate the K-theory of sheaf stacks and its functoriality properties. As applications of the localization formulas, we establish a K-theoretic wall-crossing formula for simple $\mathbb{C} ^\ast $ -wall crossings and define K-theoretic invariants refining the Jiang-Thomas virtual signed Euler characteristics.

中文翻译：

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为几乎完美的阻塞理论定位虚拟结构滑轮

作者在较早的一篇论文中引入了几乎完美的阻塞理论作为适当的概念，以定义虚拟结构滑轮和ķ- 许多感兴趣的模堆栈的理论不变量，包括ķ-Calabi-Yau 三重上滑轮和复合物的理论 Donaldson-Thomas 不变量。虚拟结构滑轮的构建是基于ķ- 捆堆的理论和 Gysin 地图。在本文中，我们将虚拟环面定位和等分定位公式及其组合推广到几乎完美的障碍理论的设置。为此，我们进一步调查ķ- 层堆理论及其功能特性。作为定位公式的应用，我们建立了一个ķ- 简单的理论穿墙公式 $\mathbb{C} ^\ast $ - 穿墙并定义ķ- 精炼江-托马斯虚符号欧拉特征的理论不变量。