Questions tagged [vanishing-cycles]

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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 $...
1 vote
0 answers
142 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", ...
2 votes
0 answers
142 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 ...
63 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 ...
4 votes
0 answers
141 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 ...
1 vote
0 answers
50 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 ...
27 votes
0 answers
921 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 (...
4 votes
1 answer
129 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 ...
0 votes
1 answer
221 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 $\...
3 votes
1 answer
260 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 ...
3 votes
0 answers
189 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 ...
4 votes
0 answers
252 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 ...
3 votes
1 answer
286 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, ...
9 votes
0 answers
309 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 ...
4 votes
0 answers
351 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 ...
6 votes
0 answers
206 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 ...
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 ...
5 votes
1 answer
554 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 votes
1 answer
490 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 $$...
4 votes
1 answer
488 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 ...
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-...
7 votes
1 answer
915 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 ...
8 votes
0 answers
316 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 ...
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 ...
3 votes
1 answer
400 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 $U$ ...