Monodromic but not equivariant sheaves and Braden's theorem

Question

Let $X$ be a complex variety with contracting $\mathbb{G}_m$ action. Let $i\colon \{x_0\}\to X$ be the inclusion of the fixed point. Then the simplest case of Braden's theorem (which would then seem to go back to Springer) says that if $\mathcal{F}\in D^b_c(X)$ is a constructible complex which is monodromic or weakly $\mathbb{G}_m$-equivariant, then $$ i^*\mathcal{F}=R\Gamma(\mathcal{F}) $$ so that the stalk at the attracting point is given by global sections.

To obtain the monodromic category, one starts with objects pulled back from the equivariant derived category via $X\to[X/\mathbb{G}_m]$, then adds all objects obtainable by finite iteration of taking cones in $D^b_c(X)$ between such already-obtained objects.

Often people apply the above theorem just to pullbacks of equivariant objects, but beyond that, are there some hypotheses on $X$ that allow easy checking of whether something is monodromic? For example, for which every complex constructible with respect to $X\setminus{x_0},~x_0$ is monodromic?

Of course, I'm also open to answers applying beyond this special case of Braden's theorem.

Answer 1

Any $\mathbb G_m$-invariant stratification such that the natural map from the fundamental group of $\mathbb G_m$ to the fundamental group of each stratum is trivial has the property that objects constructible with respect to that stratification are monodromic.

To prove this, we may clearly reduce to objects which are supported on a single stratum and concentrated in a single cohomological degree, which correspond to representations of the fundamental group of the stratum. Since these representations all factor through the fundamental group of the quotient, all these objects can be made equivariant and thus are monodromic.

I think it also can be checked that objects constructible with respect to a $\mathbb G_m$-invariant stratification satisfy $i^* \mathcal F = R\Gamma(\mathcal F)$, regardless of whether it satisfies this $\pi_1$ condition.