Let $X$ be a variety and $p\in X$ a point. Let $IC_X$ be the intersection cohomology sheaf, and let $IC_{X,p}$ be its stalk at $p$. Let $IH^*_p(X) := H^{*-\dim X}(IC_{X,p})$ be the local intersection cohomology of $X$ at $p$.
Let $TC_pX$ be the tangent cone to $X$ at $p$.
Question: What is a sufficient condition to obtain an isomorphism $IH^*_p(X) \cong IH^*_0(TC_pX)$?
If $X$ is stratified and there is an affine cone $S$ which is an etale normal slice* at $p$ to the stratum containing $p$, then $S$ has to be isomorphic to the normal cone, and the local intersection cohomology of $X$ at $p$ will be isomorphic to the local intersection cohomology of $S$ at the cone point, which will in turn be isomorphic to the local cohomology of the tangent cone at the cone point (since the tangent cone is isomorphic to the normal cone cross a vector space). So a sufficient condition for the existence of slices would also be a sufficient condition for the isomorphism.
*By "etale normal slice", I mean that there exists a vector space V and a map from an open subset of $V\times S$ to $X$, etale at $(0,0)$, taking $(0,0)$ to $p$ and $V\times\{0\}$ to the stratum. Since etale maps induce isomorphisms on tangent cones, $S$ would have to be the normal cone.
