Explicit Direct Summands in the Decomposition Theorem

Let f:X→Y be a semismall resolution of singularities. Then the pushforward of the constant sheaf on X is a semisimple perverse sheaf on Y. Under these conditions, I know how to calculate the direct summands of the pushforward f*X[dim X].

My question is as follows: What more general statements are there that enable us to explicitly calculate the direct summands of the pushforward? I'm thinking especially of the case where f:X→Y is as above (so in particular semismall), but we replace the constant sheaf ℚX with an arbitrary perverse sheaf (of geometric origin).

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Maybe this should be a separate question, but how do you calculate the direct summands in the case of the constant sheaf Q? –  Dinakar Muthiah Oct 20 '09 at 15:48
There's an answer below, but basically it's a sum over everything that prevents you from being small + Q as usual. –  Ilya Nikokoshev Oct 22 '09 at 22:27

I'm a little confused about your question. If you just want to know what the summands are there's nothing special about f*X[dim X]. The only way I know of understanding that sheaf is a general algorithm for understanding all semi-simple perverse sheaves. The only fact you use is that a simple perverse sheaf is concentrated in the highest degree allowed by perversity on the largest stratum in its support, and strictly below this degree on all smaller strata (do I need some hypothesis for this? This tends to be the sort of thing I forget).

So, if I have a semi-simple perverse sheaf F, all I have to is look at the restriction of F to each stratum. This will be a complex, whose cohomology in each term is a local system. Perversity includes an upper bound on the degrees that this cohomology can be non-zero. I take the local system in the highest degree allowed by perversity on each stratum, and take the IC sheaves of all those local systems. It happens that in the case of f*X[dim X], this local system has a geometric interpretation (it's the highest degree cohomology of the fiber allowed by semi-smallness), but that's the only thing that's special about this case.

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I want to know how to attack the problem of finding the local systems that appear, and to date I have only seen the geometric interpretation for f<sub>*</sub>&#8474;<sub>X</sub>[dim X]. Does your answer mean I can take my perverse sheaf F, and on a relevant stratum take the local system whose stalk at a point y is the highest degree cohomology of f<sup>-1</sup>(y) with coefficients in F? Also I would be interested in knowing about any good references out there for this type of material. –  Peter McNamara Oct 19 '09 at 0:17
If F is a perverse sheaf on X, then yes (assuming you are careful about what cohomology with coefficients in F means. This is just using the Cartesian diagram for the inclusion of the stratum, and the way pull-back and push-forward can go through either corner. –  Ben Webster Oct 19 '09 at 4:00
So being careful, wouldn't it be R^d f_* F on the stratum where d is the appropriate dimension? –  Peter McNamara Oct 20 '09 at 1:44

I agree with Ben that the question is a little confusing.

There are two possible questions:

How do you calculate direct summands of the direct image when we start with an IC (not necessarily the constant sheaf) on the source?

In this case I think the direct image need not be perverse. In which case you are in the general situation of describing the direct summands of a semi-simple complex, for which you need to know the characters of all IC's, or be very clever.

How do you calculate the local systems occurring in the direct image of the constant sheaf?

Ben describes what happens above. Semi-small means that the fibre over each stratum of the base has dimension bounded by half the codimension of the stratum (a number I will call d). The local system is then given by local system of 2d^th cohomology of the fibre.

In the semi-small situation this is beautiful: the 2d^th cohomology of the fibre is zero if the stratum isn't relevant, and has a basis given by fundamental classes of irreducible components if the stratum is relevant.

Note that the fundamental group of the stratum acts on the irreducible components of the fibre via monodromy, and this is precisely the local system that you get. (As an aside, this explains why the local systems are semi-simple, even though the fundamental group might be infinite: the representation always factors through the permutation group on the irreducible components.)

I first learnt this in the beautiful article "The Hard Lefschetz Theorem and the topology of semismall maps" by de Cataldo and Migliorini.

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My intention was to remain in the case of a semismall map (I've edited the question) where the direct image of a perverse sheaf is perverse. –  Peter McNamara Oct 22 '09 at 0:44
Yes, it's a wonderful article, highly recommended to everyone. –  Ilya Nikokoshev Oct 22 '09 at 22:26
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