Let $T$ be a complex torus and $X$ a smooth quasi-projective $T$-variety with finitely many fixed points. Denote by $\varphi:H_{T}(X)\rightarrow H_{T}(X^T)$ the map on equivariant cohomology induced by the inclusion $X^T\hookrightarrow X$ of the fixed points. If $X$ were compact, then we could explicitly describe $\varphi$ by computing the weights of the isotropy representations on the tangent spaces to fixed points. Still, I am wondering about what can be said if one does not assume $X$ to be compact. In particular, should one invoke essentially the same procedure to describe $\varphi$? I would appreciate references and/or suggestions.
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asked Jun 26, 2013 at 15:31
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$\begingroup$ Assuming that the action of $T$ on $X \subset \mathbb{P}(V)$ is induced from an action of $T$ on $V$, a natural thing to do would be to study $H_T(\overline{X})$ (or possibly some related complete variety), where $\overline{X}$ is the Zariski closure of $X$, using localization and then use that to recover information on $H_T(X)$. Your answer is going to depend in some way on what is going on "at infinity". $\endgroup$– Michael JoyceCommented Jun 26, 2013 at 21:11
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$\begingroup$ This is an interesting suggestion. However, $\overline{X}$, while projective and compact, will often not be smooth. In my situation, there will still be only finitely many fixed points, so are singularities an issue? $\endgroup$– Peter CrooksCommented Jun 26, 2013 at 21:59
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$\begingroup$ You are right that the singularities of $\overline{X}$ will likely play an important role. I believe there are versions of the localization theorem that apply to singular varieties, or alternatively you may try to find a $T$-equivariant resolution of $\overline{X}$. You might look at Brion's survey article "Equivariant Cohomology and Equivariant Intersection Theory", which has several versions of the localization theorem (and references to other work as well). $\endgroup$– Michael JoyceCommented Jun 26, 2013 at 22:26
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1$\begingroup$ I don't understand what "describe $\varphi$" means. Does it mean compute the kernel and cokernel? Also, since you said "complex", you're in characteristic zero and have equivariant resolutions of singularities, so may assume $\overline X$ smooth. I doubt you can expect it to have finitely many fixed points. Anyway, I would think the principal example to test intuition on is to rip the fixed points out of a projective variety. $\endgroup$– Allen KnutsonCommented Jun 27, 2013 at 1:05
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$\begingroup$ You are certainly right that "describe $\varphi$" is a bit ambiguous. For me, the objective was to write a formula for $\varphi$ along the lines of the integration of equivariant differential forms in the compact case. In terms of $\overline{X}$ having only finitely many fixed points, this holds only for my examples. You are right that it probably should not be expected in general. $\endgroup$– Peter CrooksCommented Jun 27, 2013 at 12:07
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