Let $X$ be a connected smooth algebraic variety (say over $\mathbb{C}$) and let $L$ be a local system on an open subvariety. How can we compute $Ext^i(IC(L),IC(L))$ for $i\ge 1$ ? Is there a general method or only ad-hoc arguments specific to examples?

A source of examples: let $p:X'\to X$ be a small map. Then the sheaf $\mathcal{F} = Rp_*\mathbb{C}_{X'}$ is an IC sheaf of the above form.

One could also replace cohomology with equivariant cohomology, or more generaly consider the question for algebraic stacks.

Any references where this is computed in specific examples would be also very useful.

Let $X$ be a connected smooth algebraic variety (say over $\mathbb{C}$) and let $L$ be a local system on an open subvariety. Suppose that we know $H^*(X)$. How can we compute $Ext^i(IC(L),IC(L))$ for $i\ge 1$ ? Is there a general method or only ad-hoc arguments specific to examples?

A source of examples: let $p:X'\to X$ be a small map. Then the sheaf $\mathcal{F} = Rp_*\mathbb{C}_{X'}$ is an IC sheaf of the above form. (and here suppose we also know $H^*(X')$). I would be interested to know if for this class of examples one can say something precise about the dimensions of the Ext groups.

One could also replace cohomology with equivariant cohomology, or more generaly consider the question for algebraic stacks.

Any references where this is computed in specific examples would be also very useful.