论文标题
重新审视虚拟本地化
Virtual localization revisited
论文作者
论文摘要
让$ t $是用固定基因座$ z $的代数方案$ x $作用的拆分曲线。 Edidin和Graham表明，在本地化的$ t $ equivariant chow groups，（a）沿$ i：z \ to x $沿$ i：z \ to x $的推送$ i _*$是同构的，并且（b）当$ x $平滑时，倒置$（i _*）^{ - 1} $可以通过Gysin Rualdback $ i^$ i^$ i^$ i^$ i^$ e（普通捆绑包$ n $的Euler类。在本文中，我们表明（b）使用操作的虚拟版本$ i^！$和$（ - ）\ cap e（n）^{ - 1} $时，$ x $是一种准平滑派生方案（或deligne-mumford stack）。作为推论，我们证明了虚拟本地化公式$ [x]^{vir} = i_*（[z]^{vir} \ cap e（n^{vir}）^{ - 1}）$ graber-pandharipande的$ graber-pandharipande，而没有全局分辨率，而无需全局分辨率。我们在（派生）堆栈中的固定基因座上包含一个附录，该附录应具有独立关注。
Let $T$ be a split torus acting on an algebraic scheme $X$ with fixed locus $Z$. Edidin and Graham showed that on localized $T$-equivariant Chow groups, (a) push-forward $i_*$ along $i : Z \to X$ is an isomorphism, and (b) when $X$ is smooth the inverse $(i_*)^{-1}$ can be described via Gysin pullback $i^!$ and cap product with $e(N)^{-1}$, the inverse of the Euler class of the normal bundle $N$. In this paper we show that (b) still holds when $X$ is a quasi-smooth derived scheme (or Deligne-Mumford stack), using virtual versions of the operations $i^!$ and $(-)\cap e(N)^{-1}$. As a corollary we prove the virtual localization formula $[X]^{vir} = i_* ([Z]^{vir} \cap e(N^{vir})^{-1})$ of Graber-Pandharipande without global resolution hypotheses and over arbitrary base fields. We include an appendix on fixed loci of group actions on (derived) stacks which should be of independent interest.
