Questions tagged [vanishing-cycles]
The vanishing-cycles tag has no usage guidance.
26 questions
3
votes
0
answers
100
views
Vanishing cycles and component groups
Let $A$ be an abelian variety over a local field $K$ and assume it has toric reduction. Then two classical invariants associated to this are the component group $\Phi(A)=\mathcal{A}_s/\mathcal{A}_s^0$ ...
- 4,418
1
vote
0
answers
273
views
A hard-Lefschetz theorem with torsion coefficients?
Let $X$ be a smooth projective surface over $\overline{\mathbb{F}_{q}}$. Let $\ell$ be a prime distinct from the characteristic.
Assume we have a Lefschetz pencil of hyperplane sections on $X$. Let $...
- 11
1
vote
0
answers
162
views
Definition of nearby cycle over an affine line
In some famous papers like Gaitsgory's "Construction of central elements in the affine Hecke algebra via nearby cycles" and Beilinson-Bernstein's "A proof of Jantzen conjectures", ...
- 291
4
votes
0
answers
174
views
Do the nearby cycle and Beilinson's vanishing cycle functors commute?
Let $X$ be a complex algebraic variety with a pair of regular functions $f_1,f_2$. To these functions we can associate various functors: the nearby cycles functor $\Psi_{f_i}$, the vanishing cycles ...
- 1,071
4
votes
0
answers
179
views
Is $H^*($vanishing cycles$)$ computed by the twisted de Rham complex?
In notes by Sabbah (Theorem 3), it is stated that the cohomology
$$\text{H}^*(X,\varphi_f)$$
of the vanishing cycle sheaf of a function $f:X\to \mathbf{A}^1$ for certain $X$ is expected to be the same ...
- 5,711
1
vote
0
answers
51
views
Nearby cycles morphism of Guibert-Loeser-Merle
In the paper Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink by Gil Guibert, Francois Loeser and Michel Merle, the authors defined the morphism for which I ...
- 893
4
votes
1
answer
156
views
Explicit description of perverse sheaves on a disk
In How to glue perverse sheaves Beilinson claims that the category of perverse sheaves on the complex unit disk $D$ with the stratification with the closed strata $\{0\}\subset D$ is equivalent to the ...
- 1,071
0
votes
1
answer
243
views
Nearby cycles for schemes with semi-stable reduction
Let $R$ be a henselian DVR with fraction field $K$ and residue field $k$ of characteristic $p>0$. Let $\overline K$ be an algebraic closure of $K$, $\overline R$ the normalization of $R$ in $\...
- 769
3
votes
1
answer
307
views
Are all exact Lagrangian spheres, vanishing cycles?
Let $\pi: E \to D$ be an exact Lefschetz fibration with corners (fibers with boundary)over the disk. Fix a point $\theta \in \partial D$ and consider the fiber $F_\theta = \pi^{-1}(\theta)$ over that ...
- 1,409
3
votes
0
answers
207
views
Compute the nearby cycles functor for the category of mixed motives
I am reading the survey of J. Ayoub, The motivic nearby cycles and the conservation conjecture (see here), in which he introduced the original version motivic nearby cycles (another note by Illusie is ...
- 893
4
votes
0
answers
329
views
Vanishing cycles and injectivity of the specialisation map
Consider a proper algebraic map between complex varieties $f : X \to D$ ($D$ is the unit disk), which is a submersion over $D^*$. I would like to know if they are any condition on $f$ such that the ...
9
votes
0
answers
335
views
Understanding the mixed Hodge structure on D-modules on $\mathbb{C}$
Consider the category of regular D-modules on $\mathbb{C}$ (let's say as a complex manifold). It's a well-known fact that if you consider the subcategory of these where the singular support is the ...
- 44.7k
3
votes
1
answer
312
views
Iterating specialization of sheaves?
This is a question about the operation of taking the specialization of sheaves along a subspace. I'll recall the settings in which I've encountered a notion of specialization of sheaves:
The real, ...
- 272
27
votes
0
answers
960
views
Nearby cycles without a function
Suppose that:
$X$ is a smooth complex algebraic variety,
$f : X \to D$ is a proper map to a small disc, smooth away from 0,
$Z_\epsilon = f^{-1}(\epsilon)$, and $Z = Z_0$.
Then there is a procedure (...
- 6,162
6
votes
0
answers
216
views
BRST cohomology and vanishing cycles
Consider the $\mathbb{C}$-variety $\mathbb{A}^{1}$, equipped with the potential (ie global function) $P:=\frac{z^{n+1}}{n+1}$. We can form the twisted de Rham complex $H_{dR}(\mathbb{A}^{1},P)$ which ...
user108998
5
votes
1
answer
570
views
Degeneration of smooth curves and Picard-Lefschetz formula
Let $\pi:\mathcal{C} \to \Delta$ be a family of projective curves of genus $g \ge 2$ over the unit disc $\Delta$, smooth over the punctured disc $\Delta\backslash \{0\}$ and central fiber $\pi^{-1}(0)$...
- 2,032
2
votes
1
answer
516
views
'Stalk' of vanishing cycles at $k$-point
I have a simple question on notation.
Let $S$ be a Henselian trait with closed point $s$ (with finite residue field $k$) and generic point $\eta$. Let $X/S$ be a variety. Then, we have the functor
$$...
- 787
4
votes
1
answer
511
views
Nearby cycles and specialisation - properties
I am looking for reference for properties of nearby cycles - specifically, commutation with non-characteristic pull-back (good enough - commutation with pull-back to closed subvariety which is ...
- 5,562
4
votes
0
answers
359
views
Hodge modules and Deligne-Beilinson cohomology of function fields
Let $K$ be a function field over complex numbers i.e. the fraction field of a complex variety. Then one can define the Deligne-Beilinson cohomology and mixed Hodge modules for $K$ as the direct limit ...
- 16.9k
8
votes
0
answers
318
views
The Rappoport-Zink spectral sequence vs. the one of the complement of a normal crossing divisor
As far as I understand these matters, for a regular $\mathfrak{X}$ that is proper flat of finite type over $\operatorname{Spec}\mathbb{Z}_p$, the Rappoport-Zink spectral sequence relates the etale ...
- 16.9k
7
votes
1
answer
955
views
Vanishing cycles of a locally constant sheaf for a smooth morphism in the $l = p$-case
$\DeclareMathOperator{\Spec}{Spec}$
My question is concerned with vanishing cycles of a locally constant sheaf for a smooth morphism in the case $l = p$. In the case $l \neq p$ this is a statement in ...
- 146
3
votes
1
answer
418
views
How to glue perverse sheaves of abelian groups?
Let $X$ be a complex algebraic variety and consider the category $P(X)$ of perverse sheaves of complex vector spaces.
Let $f:X\rightarrow \mathbb C$ be a regular function, $Z$ its zero set and $U$ ...
- 13.2k
34
votes
3
answers
7k
views
Vanishing cycles in a nutshell?
To quote one source among many, "the general reference for vanishing cycles is [SGA 7] XIII and XV". Is there a more direct way to learn the main principles of this theory (i.e. without the language ...
- 1,141
10
votes
1
answer
1k
views
Computation of vanishing cycles
Here's the problem I'm looking at:
$F$ is a perverse sheaf (or a regular holonomic D-module, or even a mixed hodge module) on $\mathbb{C}^2$ stratified by $z_1 = 0$, $z_2=0$. It can be caracterized ...
- 7,527
66
votes
4
answers
11k
views
Is there a good way to think of vanishing cycles and nearby cycles?
Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at ...
- 45.7k
6
votes
2
answers
1k
views
Higher vanishing cycles
The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned a connection with Bousfield localization. I find the Morava's remarks un-...
- 10.8k