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Could you give me an example of a clear and beautiful application of Bott residue formula in torus-equivariant cohomology (see below)?

I found an example calculating a product of Chern classes on singular quadric in section 5.3 of Edidin, Graham Localization in equivariant intersection theory and the Bott residue formula, but in singular case one need equivariant Chow groups instead of equivariant cohomology, and anyway the example is too technical.


Let $T=(\mathbb C^*)^n$ be an algebraic torus acting on a smooth complex projective variety $X$ such that there is a finite number of $T$-fixed points of $X$. A technically simple case of Bott residue formula says that for $T$-equivariant vector bundle $E$ with $\operatorname{rk} E=\dim X$ the top Chern class is a number given by $$\int_X c_{top}(X, E)=\sum_{x \in X^T} \frac{c_{top}^T (x, E_x)}{c_{top}^T (x, TX_x)},$$ where

  • the right hand side should be calculated in the quotient field of $T$-equivariant cohomology ring of a point, that is simply $$\operatorname{Quot} H_T^*(x)=\operatorname{Quot} \mathbb Q[t_1, \ldots, t_n]=\mathbb Q(t_1, \ldots, t_n);$$

  • the $T$-equivariant top Chern class $c_{top}^T(x, V)$ of a $T$-representation $V$ (above it is the $x$-fiber $E_x$ or $TX_x$) is the product of $T$-characters of $V$ as elements $$a_1t_1+\ldots+a_nt_n \in \mathbb Q[t_1, \ldots, t_n];$$

  • the equality is taken in $\mathbb Q(t_1,\ldots,t_n)$ as the summands lie there, but the sum lies in $\mathbb Q$.

So one can derive the formula $\chi(X)=\chi(X^T)$ by choosing $E=TX$, but it is too degenerate. Could you give me an example which is more meaningful but still clear enough? Any reference or idea is also welcome!

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  • $\begingroup$ this paper looks interesting arxiv.org/pdf/1610.02539v1.pdf but I still have to study it myself. They apply Atiyah-Bott formula for the problems of additive combinatorics. $\endgroup$ Nov 13 '16 at 19:12
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Consider the line bundle $E = \mathcal{O}(n)$ on $X = \mathbb{CP}^1$ with the standard action of $\mathbb{C}^*$ by rotation. The fixed points are $0$ and $\infty$, and the fixed point formula involves a non-trivial cancellation.

This example is somewhat more interesting in equivariant K-theory, where it's computing the character of the representation of $SL_2$ on the global sections.

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There is the famous paper by Ellingsrud and Strømme, who apply Bott's residue formula to enumerative geometry:

Geir Ellingsrud and Stein Arild Strømme. Bott's formula and enumerative geometry. J. Amer. Math. Soc. 9 (1996), 175-193

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