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The Atiyah-Bott-Berline-Vergne-Witten localization formula says $S^1$ acting on compact manifold $M$ isolated fixing points. And for a closed equivariant form $\omega$, then $$(2\pi)^{-\frac{\dim(M)}{2}}\int_M\omega=\sum_{p\in isolated~ point}\frac{\omega^0(p)}{\det^{1/2}(L_p)},$$ where $L_p$ denotes the induced action on $TM$.

Q:

  • If the isolated-point set is a high dimensional submanifold, are there some results about the Berline-Vergne?

  • In the research of the localization of equivariant differential forms, are there some open problems?

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    $\begingroup$ Theorem 4.1 mi.uni-koeln.de/~dgreb/localisation.pdf $\endgroup$
    – user21574
    Commented Jan 28, 2017 at 3:19
  • $\begingroup$ See paper of Atyah and Bott in 82 $\endgroup$
    – user21574
    Commented Jan 28, 2017 at 3:23
  • $\begingroup$ Yes. In the Duistermaat and Heckman's second paper, they generalized the DH-localization formula to the high dimensional case. This could answer the second question. $\endgroup$
    – DLIN
    Commented Jan 28, 2017 at 5:18
  • $\begingroup$ @DLN, I'm very interested in your second question: what are open questions about localization theorem in equivariant cohomlogy, have you got answer for this question ? $\endgroup$
    – Mira
    Commented Dec 11, 2021 at 23:58

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