Let $X$ be a smooth complex projective variety acted upon by algebraically by a complex torus $T$. Let $F_1,\ldots,F_n$ be the connected components of $X^T$ and assume that the restriction map $$H_T^*(X)\rightarrow H_T^*(X^T)=\bigoplus_{i=1}^nH_T^*(F_i)$$ is injective. (For me, all equivariant cohomology is over $\mathbb{Q}$.) For each $i$, let $N_i$ denote the normal bundle of $F_i$ in $X$, and let $e_T(N_i)\in H_T^*(F_i)$ denote its $T$-equivariant Euler class.

$\textbf{Question}:$ Viewing $H_T^*(X)$ as a subalgebra of $\bigoplus_{i=1}^nH_T^*(F_i)$, when is $H_T^*(X)$ generated as an algebra by the equivariant Euler classes $e_T(F_i)$?