We show that a perfect obstruction theory for a $\mathbb{G}_\text{m}$-gerbe determines a semi-perfect obstruction theory for its base, which is perfect if the gerbe is quasi-compact and affine-pointed. These results streamline the construction of a semi-perfect obstruction theory for the base, and allow us to relate virtual classes of the gerbe and its base.

中文翻译：

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$$\mathbb {G}_\text {m}$$-gerbes 的虚拟类

我们证明了 $\mathbb{G}_\text{m}$-gerbe 的完美阻塞理论确定了其基的半完美阻塞理论，如果 gerbe 是准紧致和仿射点，则这是完美的。这些结果简化了基部半完美阻塞理论的构建，并使我们能够将 gerbe 的虚拟类与其基部联系起来。