extensions of IC sheaves

Question

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.

Answer 1

Not sure what to say in general, but in practice one often uses a semismall resolution. For example if your sheaf $IC(L)$ is given as $\pi_* C_Z$ (pushforward of constant sheaf - I'm assuming $Z$ smooth and oriented) for a proper $Z\to X$ you can calculate derived self-maps $$Hom(\pi_*C_Z,\pi_*C_Z)= Hom(C_Z, \pi^!\pi_*C_Z).$$ Now write $\pi_1,\pi_2$ for the two projections from $Z\times_X Z\to Z$. Then we calculate further

$$Hom(C_Z, \pi^!\pi_*C_Z)=R\Gamma_Z(\pi_{1,*}\pi_2^!C_Z)=R\Gamma_{Z\times_X Z}(\omega_{Z\times_X Z}),$$

i.e., Borel-Moore homology of $Z\times_X Z$ (up to a shift I've ignored). This works in the equivariant setting (i.e. for sheaves on stacks) equally well, giving equivariant Borel-Moore homology. The famous example of this is the case where $X$ is the nilpotent cone [or its quotient by the group], $Z$ the [equivariant] Springer resolution, and $Z\times_X Z$ the [equivariant] Steinberg variety, cf. the book of Chriss-Ginzburg. In this case the equivariant BM homology (=Ext algebra of the Springer sheaf) is the degenerate affine Hecke algebra. This calculation works as is in the dg setting --- for the statement on the level of derived categories you need formality, which in the Springer case is a result of Laura Rider here.

Answer 2

There is general convolution algebra type formalism that you can try and use, see Chriss and Ginzburg's "Complex geometry and representation theory". Chapter 8 in particular is very much in line with the your "source of examples".

In general this can be hard. Heck, consider even the case that your local system is trivial and your IC-sheaf is the (shifted) constant sheaf on $X$. Then you are asking to compute the cohomology of the space. This may be a non-trivial endeavor depending on your space.

An ideal example where things work out very nicely is that of flag varieties and the IC-sheaves are those corresponding to Schubert subvarieties. Then these Ext-computations can be carried out combinatorially in the Hecke algebra. Soergel's papers on this and related topics are particularly enlightening. A related point here is that in this situation considering hypercohomology as a functor to graded modules for the cohomology algebra of the flag variety is full and faithful. This result also generalizes to projective varieties with $\mathbb{C}^*$-actions. This is a result of Ginzburg "Perverse sheaves and $\mathbb{C}^*$-actions". Similar ideas are also worked out in some of Springer's papers on spherical varieties. Related are also the moment graph techniques that can be found in papers of Braden, MacPherson, etc.

Regarding, "replace cohomology with equivariant cohomology". I am assuming you want to compute $Ext$ in the equivariant derived category. Then similar techniques as above can be tried. In the presence of suitable assumptions, often the equivariant calculation reduces to the non-equivariant one due to formality. Instead of trying to flesh this out let me just refer to Soergel's "Langlands philosophy and Koszul duality". A number of examples are worked out in there.

In general though, there is no magic pill that I know of.