# Monodromic but not equivariant sheaves and Braden's theorem

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.

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.
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.