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Is there a written version of Atiyah-Bott localization formula for varieties/manifolds with toric singularities? More precisely, suppose $F$ is a fixed locus of a torus action $\mathbb{T}={\mathbb{C}^*}^m$ on $X$, that is smooth itself but has a normal cone $N$ whose fibers are modeled on an affine toric variety $Y_{\sigma}$ given by some cone $\sigma\subset \mathbb{R}^k$. In the smooth case $Y_\sigma\cong \mathbb{C}^k$, $N\to F$ is a rank $k$ vector bundle, and the localization contributaion of the fixed locus $F$ is $$ 1/e_{\mathbb{T}}(N), $$ where $e_{\mathbb{T}}(N)$ is the equivariant Euler class of the normal bundle $N$.

What is this factor when $Y_{\sigma}$ is singular?

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  • $\begingroup$ Hello Mohammad, I hope that you are doing well. Are you consindering only finite quotient singularities that arise on a toric variety (i.e., Abelian finite quotient singularities)? Or are you considering more general toric singularities. For "orbifolds", you can consult Melissa Liu's contribution to the Handbook of Moduli. $\endgroup$
    – Jason Starr
    Commented Sep 1, 2019 at 10:14
  • $\begingroup$ Hi Jason, thanks. Not just orbifolds. I want to use localization on moduli spaces of log curves. Potentially, any affine toric variety $Y_{\sigma}$ (corresponding to some rational convex cone in $\mathbb{R}^k$) can arise as the normal cone of some stratum. Orbifold case does not suffice. $\endgroup$
    – Mohammad Farajzadeh-Tehrani
    Commented Sep 3, 2019 at 14:41

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In general, the factor you are asking about is often called the equivariant multiplicity of $X$ along the fixed component $F$. The best source for this is Brion's paper "Equivariant Chow groups for torus actions" (Transformation Groups, 1997), available here:

https://www-fourier.ujf-grenoble.fr/sites/default/files/ref_365.pdf

(Look at Section 4, and also at Section 5 which gives formulas for toric singularities.) There is also this paper, https://arxiv.org/abs/1907.00076, which puts things in a more general context and makes the connection with the Atiyah-Bott formula a bit more explicit.

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  • $\begingroup$ Thanks for the references. I will go through these. I wonder how much of these can be generalized to the original differential geometry setup of Morse-Bott by allowing manifolds with singularities modeled on toric varieties. $\endgroup$
    – Mohammad Farajzadeh-Tehrani
    Commented Aug 30, 2019 at 19:54

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