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?