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?

• For other formulas, e.g. Duistermaat-Heckman formula, are there some results for the high dimensional isolated-point set?