In this post I’ll talk about a really fun trick Bhargav explained to me yesterday.
Let be a field with separable closure . Algebraic variety over means separated -scheme of finite type. Let be a prime invertible in . Perverse sheaf means perverse -sheaf.
If is a proper map of algebraic varieties over , the decomposition theorem tells you that after base extension to there is a direct sum decomposition
in . Here is some finite set of closed subvarieties, and is a lisse -sheaf on the smooth locus of . (My convention is that where is the natural map, so . Some people have different conventions for shifts here.)
The decomposition is non-canonical. In particular, it is not -equivariant, so it has no reason to descend to an analogous direct sum decomposition of in . Indeed, typically there is no such decomposition! However, as Bhargav explained to me, one can still descend a certain piece of to when is projective. This turns out to be good enough for some interesting applications.
To present Bhargav’s trick, let me make a definition. (What follows is a slight reinterpretation of what Bhargav told me, all mistakes are entirely due to me.)
Definition. Let be a perverse sheaf on a variety . Let be the inclusion of the maximal dense open subvariety such that is a (shifted) lisse sheaf. Then we define the generic part of as the perverse sheaf .
Note that is zero if and only if is supported on a nowhere-dense closed subvariety. Also, in general there is no map between and . However, in some cases is a direct summand of :
Proposition. Let be a perverse sheaf on a -variety , and suppose that the pullback of to is a direct sum of IC sheaves. Then is a direct summand of .
Proof. Let be as in the definition of the generic part of , with closed complement . Our assumptions together with the definition of the generic part guarantee that for some closed subvarieties contained in .
Now look at the natural maps . Set and . Since does not admit any nonzero quotient supported on , the composite map is zero. Thus factors over an inclusion . Moreover, has the same generic part as . This is enough to imply that , so we have a natural inclusion . A dual argument shows that factors over a surjection . It is now easy to see that the composite map is an isomorphism, so is a direct summand of .
Corollary 0. Let be a projective map of -varieties. Then is a direct summand of .
Proof. The decomposition theorem and the relative hard Lefschetz theorem give a decomposition in . Then is a direct sum of IC sheaves after pullback to , so we can apply the previous proposition.
Corollary 1. Let be a projective alteration of -varieties with smooth. Then is a direct summand of .
Proof. Check that is a direct summand of by playing with trace maps. Now apply the previous corollary.
Corollary 2. Let be a finite extension. Then for any -variety , the -adic intersection cohomology is a de Rham -representation.
Proof. Let be a resolution of singularities. The previous corollary shows that is a direct summand of compatibly with the -actions. Since is de Rham and the de Rham condition is stable under passing to summands, we get the result.
Note that we can’t prove this corollary by applying the decomposition theorem directly out of the box: the decomposition theorem does immediately give you a split injection , but this map is not guaranteed a priori to be -equivariant!
Corollary 3. Let be a finite extension of or . If satisfies the weight-monodromy conjecture for all smooth projective -varieties , then satisfies the weight-monodromy conjecture for all proper -varieties . In particular, the weight-monodromy conjecture holds for the -adic intersection cohomology of all proper -varieties for finite.
Proof. Entirely analogous to the previous proof.
It would be interesting to know whether Corollary 1 has a “motivic” incarnation. Here I will pretend to understand motives for a minute. Suppose we have an assignment on quasi-projective -varieties, where is a suitable triangulated category of constructible motivic sheaves on with -coefficients. This should come with the formalism of (at least) the four operations , and with faithful exact -adic realization functors compatible with the four operations. I think this has all been constructed by Ayoub, maybe with some tiny additional hypothesis on ? Let denote the symmetric monoidal unit. It then makes sense to ask:
Question. In the setting of Corollary 1, is there an idempotent such that for all ?
This would imply that the split injections provided by Corollary 1 can be chosen “independently of ”, i.e. that they are the -adic realizations of some split injection in .