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Results tagged with perverse-sheaves
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user 4144
15
votes
Accepted
What exactly does the weight filtration in Hodge theory have to do with the Weil conjectures?
Dear Akhil,
This is a big topic, although one that has been discussed at various times here, e.g.
In what setting does one usually define mixed sheaves and weights for them?
The idea is that for co …
- 35.2k
6
votes
Accepted
Polarizable variations of (mixed) Hodge structures
You have my sympathies for trying come to grips with this stuff. Fortunately the answer, which is both yes and no, can be explained
in the simplest case of a point. A polarization on a pure Hodge st …
- 35.2k
6
votes
In what setting does one usually define mixed sheaves and weights for them?
I'm not sure if this is the kind of answer that you're looking for. This is an extension of Tom Nevin's answer. Saito's theory of mixed Hodge modules is modeled, to some extent, on BBD. The results ar …
- 35.2k
5
votes
Accepted
Easy special cases of the decomposition theorem?
Well, it depends on what you mean by "easy". A special case, which I find very instructive, is a theorem of Deligne from the late 1960's.
Theorem. $\mathbb{R} f_*\mathbb{Q}\cong \bigoplus_i R^if_*\ma …
- 35.2k
6
votes
What's an example of an intersection cohomology sheaf whose stalks are pure but not pointwis...
This isn't really a bona fide answer. It's more a series of thoughts, which perhaps
you or someone else can complete.
Take a normal surface singularity $(X,x)$, whose resolution (which exists in any
…
- 35.2k
6
votes
Singular support of an irreducible perverse sheaf
My impression is that the singular support of a perverse sheaf is the characteristic variety of the regular holonomic $D$-module corresponding to it under Riemann-Hilbert. Assuming that's the case, it …
- 35.2k
28
votes
Accepted
A nice explanation of what is a smooth (l-adic) sheaf?
I'll give an answer, only because I'm interested in chasing down these references
myself. But all I'm doing is assembling references. I assume that
BCnrd will keep me honest. [July 21: I've added som …
- 35.2k
30
votes
Accepted
Gabber's original proof of his purity theorem
It's different, but it also uses Weil II. See
Purity for intersection cohomology after Deligne-Gabber
for my translation of the original.
- 35.2k
6
votes
Applications for intersection (co)homology and for the Decomposition Theorem for students?
Mikhail,
Coincidentally, I am running a seminar on some of this as well, although I may be assuming a bit more background. But here is an idea which may be suitable for your students. Suppose that $ …
- 35.2k
4
votes
Accepted
Example of an intersection complex not concentrated in a single degree
Sorry I haven't read your entire question, which is a bit long. This is really just an extended comment to address the "where I should look next?" part. Suppose $X$ has an isolated singularity $x$, …
- 35.2k