This semester in Berkeley, we have been running a learning seminar on Weil conjecture and perverse sheaves following the book by Kiehl and Weissauer. All of this business with “weights” used to scare me a great deal, but after this semester, I feel I understand a bit. I thought I would try to share with everyone a bit of what we have been learning. It will also give me an opportunity to make sure that I understand this stuff.
Let denote a variety over a finite field and let denote its base change to the algebraic closure.
The first important point is that if X is such a variety, then there is a well behaved etale cohomology of X, . This cohomology behaves like you would expect it too — in particular, if there is an obvious version of X over the complex numbers, then typically this etale cohomology will agree with the ordinary cohomology of the complex points of X. The development of etale cohomology is actually quite complicated and was not the subject of our seminar and will not be the subject of this post.
Since comes from by base change, comes with a Frobenius map . For example if , then . Now, this Frobenius map gives us a endomorphism of the etale cohomology .
If is smooth and proper, then the following amazing theorem is true: The eigenvalues of acting on are all algebraic integers of size (this is the “Riemann hypothesis” part of the Weil conjecture).
(There are some notes online of a talk by Beilinson which give some complex geometry motivation for this statement.)
Let us abstract this notion. If is a vector space with an endomorphism , then we say that is pure of weight w if all eigenvalues of have size . We say that V has weight if all eigenvalues of F have size . So if X is proper, then is pure of weight i.
Now, we consider more general constructible sheaves on , which you can think of as coming from gluing local system on subvarieties. If A is such a sheaf, and , then F acts on the stalk . We say that A has weight , if all stalks of have weight . Then the above theorem generalizes as follows: If is a morphism, and the weight of A is less than w, then the weight of is also less than .
(Actually I’ve overly simplified this definition of weight. You have to check not just the points, but also the points over all and the corresponding Frobenii.)
Let us now generalize some more. We consider complexes of sheaves A on . We say that A has weight , if each has weight . We say that A has weight , if has weight (here D denote Verdier duality). We say that A is pure if it has weight and weight .
The Riemann hypothesis part of the Weil conjecture generalizes as: if is a morphism and A has weight , then has weight . In particular, using duality we see that if f is proper and A is pure of weight w, then is pure of weight w.
Now we move to perverse sheaves, which form an interesting abelian subcategory of the derived category of constructible sheaves, closed under Verdier dual. Perverse sheaves interact nicely with weights. Here are two lemmas which illustrate this interaction and then the two main theorems of this post.
Lemma Every simple perverse sheaf is pure.
Lemma If A and B are pure perverse sheaves with wt(A) < wt(B), then there are no non-trivial extensions of A by B.
Theorem (Weight filtration) Any perverse sheaf has a canonical weight filtration. This is a filtration by subobjects such that all subquotients are pure of increasing weight.
Theorem (Decomposition) Any pure perverse sheaf on X is isomorphic to a direct sum of simple perverse sheaves.
Let me illustrate the weight filtration and decomposition theorems with a couple of examples.
If f is the inclusion of into , then has the following weight filtration . We have that is pure of weight 1 and the quotient is pure of weight 2 (here is the skyscraper sheaf at , and denotes Tate twist).
Generalizing this example, we may more generally consider the inclusion of an open subset U into a smooth variety whose complement is a divisor. I’ve heard that the corresponding weight filtration on can be “seen” for varieties over using Deligne’s log complex. Does anyone have a reference for that?
Consider the map . Here, denotes the nilpotent cone for the Lie algebra sl(2). Then f is proper and semismall and so is a pure perverse sheaf on and hence by the decomposition theorem it is a direct sum of simple perverse sheaves. In fact .
Note for experts: to simplify things, throughout this post I’ve fixed an isomorphism between and and I’ve assumed that all sheaves are mixed.