Localized torus-equivariant Borel–Moore classes

Assume the Axiom of Choice. Let $r\ge 0$, let $T=({\mathbb{C}}^{\times }{)}^r$ act algebraically on a separated finite-type complex variety $X$, and suppose $X$ has a finite cover by $T$-invariant affine open subvarieties. Use rational coefficients. Let $K=(S^1{)}^r$ be the maximal compact subgroup of $T$, and use the product right Milnor model $E_K=(E_{S^1}{)}^r$, $B_K=E_K\mathrm{/}K=({\mathbb{C}\mathbb{P}}^{\mathrm{\infty }}{)}^r$. The inclusion $K\subset T$ gives the same topological Borel construction up to homotopy. For each algebraic character $\chi :T\to {\mathbb{C}}^{\times }$, let $L_{\chi }=E_K{\times }_K{\mathbb{C}}_{\chi {\mathrm{\mid }}_K}\to B_K$, with right equivalence $(e,z)\cdot t=(e\cdot t,\chi (t{)}^{-1}z)$, and put $u_{\chi }=c_1(L_{\chi })$. For the coordinate characters ${\chi }_i$ and $u_i=u_{{\chi }_i}$, the coefficient ring is $R_T=H^{\ast }(B_K;\mathbb{Q})=\mathbb{Q}[u_1,\dots ,u_r]$, with $\mathrm{\mid }{u}_i\mathrm{\mid }=2$ and $u_{\chi }={\sum }_i{n}_i{u}_i$ for $\chi (t_1,\dots ,t_r)={\prod }_i{t}_i^{n_i}$. Let $S_T$ be the multiplicative set consisting of $1$ and finite products of $u_{\chi }$ for nontrivial $\chi$.

Let $H_j^{\mathrm{B}\mathrm{M},T}(X;\mathbb{Q})={\mathrm{Hom}}_{D_{T,c}^b(X;\mathbb{Q})}({\mathbb{Q}}_X,{\omega }_X^T[-j])$, where ${\omega }_X^T$ is the normalized equivariant dualizing object whose restriction is the ordinary dualizing object ${\omega }_X$, without a quotient-stack dimension shift. Equivalently, this is the stable value in degree $j$ of $H_{j+2rm}^{\mathrm{B}\mathrm{M}}(X_m;\mathbb{Q})$, where $U_m=({\mathbb{C}}^{m+1}\setminus \{0\}{)}^r$ has the free right scalar $T$-action, $X_m=(U_m\times X)\mathrm{/}T$ uses $(e,x)\cdot t=(e\cdot t,t^{-1}\cdot x)$, and the supplied bounded-Hom comparison is taken in its stable range. Put $H_{\ast }^{\mathrm{B}\mathrm{M},T}(X;\mathbb{Q})={⨁}_{j\in \mathbb{Z}}{H}_j^{\mathrm{B}\mathrm{M},T}(X;\mathbb{Q})$. Pullback from $B_K$ and cap product make this total group an $R_T$-module: a homogeneous $a\in R_T^q$ sends $H_j$ to $H_{j-q}$, so coefficients have homological degree $-q$. For an $R_T$-module $M$, $S_T^{-1}M$ consists of fractions $m\mathrm{/}s$ with $m\in M$, $s\in S_T$, and $m\mathrm{/}s=m^{\mathrm{\prime }}\mathrm{/}{s}^{\mathrm{\prime }}$ exactly when some $t\in S_T$ satisfies $t(s^{\mathrm{\prime }}m-s{m}^{\mathrm{\prime }})=0$; addition and scalar multiplication are the induced fraction operations. Localize the total graded module and define

$${\mathcal{H}}_j^T(X):={(S_T^{-1}{H}_{\ast }^{\mathrm{B}\mathrm{M},T}(X;\mathbb{Q}))}_j\mathrm{.}$$

For homogeneous $\beta \in H_k^{\mathrm{B}\mathrm{M},T}(X;\mathbb{Q})$ and $s\in S_T$ of cohomological degree $q$, the fraction $\beta \mathrm{/}s$ has homological degree $k+q$. The notation $S_T^{-1}{H}_j^{\mathrm{B}\mathrm{M},T}(X;\mathbb{Q})$ will mean ${\mathcal{H}}_j^T(X)$, never localization of the individual group $H_j$. Likewise localized equivariant cohomology means localization of ${⨁}_q{H}_T^q(X;\mathbb{Q})$ with $\mathrm{deg}(\alpha \mathrm{/}s)=\mathrm{deg}\alpha -\mathrm{deg}s$. Cap product extends by $(\alpha \mathrm{/}s)\cap (\beta \mathrm{/}t)=(\alpha \cap \beta )\mathrm{/}(st)$. At a point, $H_j^{\mathrm{B}\mathrm{M},T}(\mathrm{p}\mathrm{t};\mathbb{Q})=H^{-j}(B_K;\mathbb{Q})$, so ${\mathcal{H}}_j^T(\mathrm{p}\mathrm{t})=(S_T^{-1}{R}_T{)}^{-j}$. Proper equivariant maps induce degree-preserving pushforwards, invariant open inclusions induce Borel–Moore restrictions, and a smooth pure $d$-dimensional complex variety has its complex-oriented fundamental class $[X{]}_T$ in degree $2d$. For $r=0$, $R_T=\mathbb{Q}$ and $S_T=\{1\}$.

[Equivariant Borel–Moore homology and normalized dualizing object] Ginzburg, Section 1, Borel–Moore homology

[Algebraic torus localization and smooth fixed-locus denominator] Edidin–Graham, Theorems 1–2