All Questions
15
questions
1
vote
1
answer
341
views
Tensor product and semisimplicity of perverse sheaves
Let $X/\mathbb{C}$ be a smooth algebraic variety. Let $D_c^b(X,\bar{\mathbb{Q}}_{\ell})$ be the category defined in 2.2.18, p.74 of "Faisceaux pervers" (by Beilinson, Bernstein and Deligne). ...
- 545
8
votes
0
answers
331
views
Beilinson's theorem for fixed stratifications
Beilinson's theorem states that for a variety $X$ and a field $k$ the realization functor
$$\text{real}: D^b\text{Perv}(X,k)\to D_c^b(X,k)$$ is an equivalence of categories.
If we only consider ...
- 1,061
6
votes
1
answer
311
views
Understanding an involution of the category of perverse sheaves on $\mathbb{C}$
It is well-known (for example: chapter 2 in [GGM] A. Galligo, M. Granger, P. Maisonobe. D-modules et faisceaux pervers dont
le support singulier est un croisement normal. Ann. Inst. Fourier Grenoble,
...
- 375
4
votes
1
answer
438
views
Perverse sheaves on the complex affine line
Show that a perverse sheaf on $\mathbb{A}^1(\mathbb{C})$ (the complex plane with the analytic topology) is a bounded complex $A$ of sheaves of $\mathbb{Q}$-vector spaces with constructible cohomology ...
- 1,479
2
votes
0
answers
173
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Perverse sheaves and maximal genus Gopakumar-Vafa invariants
Let $f: X \to Y$ be a proper morphism between complex varieties (the varieties as well as the map may be non-smooth) and let $\phi \in \text{Perv}(X)$ be a perverse sheaf on $X$. Given this data, it ...
- 1,701
3
votes
0
answers
393
views
Stalks of perverse cohomology sheaves?
For a complex of sheaves $\cal{F}^{\bullet}$ on a variety $X$, a useful fact is that the stalks of the cohomology sheaves of $\mathcal{F}^{\bullet}$ agree with the cohomology groups of the complex of ...
- 1,701
1
vote
0
answers
165
views
Espace étalé for derived category
It is known that for a sheaf $\mathcal{F}$ on $X$, we can associate $X_\mathcal{F}$, the étalé space of $\mathcal{F}$ over $X$ such that section of $X_\mathcal{F}$ coincides with section of $\mathcal{...
- 677
4
votes
1
answer
272
views
Convolution of $\ell$-adic sheaves is commutative if the group is commutative
[This is a duplicate of this question on Stackexchange]
I am trying to figure out how to prove a very basic statement about convolution of $\ell$-adic/perverse sheaves in Katz's "Rigid local systems" ...
- 329
5
votes
1
answer
349
views
Base change and the octahedron axiom
I am trying to understand "de Cataldo, Migliorini. The perverse filtration and the Lefschetz hyperplane theorem. Annals of Mathematics, 171(2010), 2089-2113." My question is about one detail in the ...
- 143
0
votes
0
answers
330
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on the Springer sheaf
Let $\pi:\tilde{\mathfrak{g}}\rightarrow \mathfrak{g}$, be the Grothendieck-Springer resolution, where $\mathfrak{g}$ is a semisimple Lie algebra over $\mathbb{C}$.
We know that $\pi$ is small thus $\...
- 3,452
1
vote
3
answers
1k
views
Perverse sheaves and tensor product
If $X$ is a connected algebraic variety of finite type over $k$ (with $k$ a field of positive characteristic) of dimension $d$, and if $\mathcal{F}$ and $\mathcal{G}$ are perverse sheaves on $X$ so $(\...
- 335
4
votes
1
answer
366
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flat descent for perverse sheaves
Let $E \in D^{b}_{c}(X,\overline{\mathbb{Q}}_{l})$ where $X$ is a $k$ scheme of finite type for a field $k$.
Let $Y\rightarrow X$ a finite flat surjective morphism such that $f^{*}E$ is perverse and ...
- 3,452
12
votes
1
answer
1k
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On the derived category of constructible étale sheaves
The derived category $D^{\flat}_{c}(X,R)$ of constructible sheaves of $R$-modules on $X_{et}$ is defined as the full subcategory of $D^b(X,R)$ whose cohomology sheaves are all constructible.
Clearly, ...
- 15.5k
8
votes
1
answer
932
views
l-adic vs complex Perverse Sheaves
Let $X$ be a scheme of finite type over $Spec(\mathbb{C})$. Let $X_{an}$ denote the associated complex analytic space. After fixing an isomorphism $\overline{\mathbb{Q}}_l\cong \mathbb{C}$, by $\S$6....
- 491
22
votes
4
answers
4k
views
Examples for Decomposition Theorem
There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne.
Here's a refresher: by $IC$ one means the intersection complex, which is just $\mathbb Q$ for a smooth ...
- 15k