The **Berline-Vergne Theorem** 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?