# Tagged Questions

**4**

votes

**0**answers

71 views

### a technical question on the definition of connections with regular singularities

Let $X$ be a quasi-projective variety over a field $k$ of characteristic zero. A good compactification of $X$ means a projective variety $\overline{X}$ containing $X$ as the complement of a simple ...

**8**

votes

**1**answer

178 views

### regular singularities and comparison isomorphism

Let $X$ be a smooth variety over some field $k \subset \mathbb{C}$. By a theorem of Grothendieck, one has a canonical isomorphism of complex vector spaces
$$
\mathbb{H}^j(X, \Omega_{X/k}^\bullet) ...

**4**

votes

**1**answer

179 views

### how is the dual connection defined?

Let $E$ be a vector bundle (i.e. locally free $\mathcal{O}_X$-module) on some smooth algebraic variety $X$ and let $\nabla: E \to E \otimes \Omega^1_X$ be an integrable connection.
I have seen that ...

**3**

votes

**2**answers

177 views

### Kernel of an integrable holomorphic dee-bar connection is a holomorphic vector bundle

I'm looking for references for the following fact, to pass on to a grad student. I think I can prove it, but it is a bit sloppy and I'd rather have something which is already written up.
Let $X$ be a ...

**2**

votes

**1**answer

188 views

### comparison theorem for connections with regular singularities

This is something I've never understood.
Let me first recall the classical case, in which you start with a smooth algebraic variety $X$ over $\mathbb{C}$. One has the algebraic de Rham complex ...

**4**

votes

**1**answer

189 views

### locally free sheaves and logarithmic connections

I know that, when $\mathcal{F}$ is a coherent sheaf on a smooth algebraic variety $X$ over a field $k$ equipped with a connection
$$
\nabla: \mathcal{F} \to \mathcal{F} \otimes \Omega^1_X,
$$ then ...

**4**

votes

**1**answer

149 views

### Chern class of a logarithmic connection

Let $X$ be a smooth complex projective algebraic variety and $E$ a line bundle on $X$. It is a classical result that if $E$ carries an integrable connection, then the first Chern class $c_1(E)$ ...

**2**

votes

**0**answers

131 views

### Lefschetz hyperplane section theorem for connections

Let $X$ be a projective, smooth, algebraic variety over a subfield of the complex numbers, and let $Y \hookrightarrow X$ be a smooth hyperplane section of $X$. The classical Lefschetz theorem claims ...

**7**

votes

**2**answers

513 views

### How many flat connections has a line bundle in algebraic geometry?

Suppose $X$ is a projective variety over $\mathbb C$. I am happy to entertain more or different adjectives — I'm not looking for the most general statement, but rather to understand when and how ...

**1**

vote

**1**answer

193 views

### connections with regular singularities

Let $k$ be a field of characteristic zero, $X=\mathbb{G}_{m, k}=\mathrm{Spec}\ k[t, t^{-1}]$ the multiplicative group over k and $E=\mathcal{O}_X$ the trivial line bundle.
Consider the connection ...

**0**

votes

**1**answer

177 views

### regular singularities

Hi friends,
Let me ask you about connexions having regular singularities. So imagine $X$ is some smooth algebraic variety over a subfield $k \subset \mathbb{C}$ and you have a locally free ...

**2**

votes

**1**answer

288 views

### cohomology of the Gauss-Manin connection

Let $U$ be a smooth algebraic variety defined over $k \hookrightarrow \mathbb{C}$. Let $\mathcal{E}$ be a locally free sheaf on $U$ equipped with an integrable connection
$\nabla: \mathcal{E} \to ...

**5**

votes

**0**answers

176 views

### Does every stack with a connection admit an atlas with a connection?

Dear all,
Let $S$ be a scheme in characteristic $0$,
and let $\mathscr{X}/S$ mean a crystal in Artin stacks over $S$ in the sense of this handout, page 4, Definition 0.5, where we replace the scheme ...

**11**

votes

**2**answers

1k views

### The algebraic Version of Riemann Hilbert Correspondence

It is well known that if I have a differentialable manifold (holomorphic maniford) $M$, then I have a functor from the categroy of vector bundles on $M$ with flat connections to the categroy of local ...

**5**

votes

**3**answers

771 views

### For quasi-coherent D-Modules

It is well-know that the category of coherent D-modules over a smooth algebraic $k$-scheme is a Tannakian category. So it is equivalent to the category of finite representations of some affine group ...

**5**

votes

**1**answer

549 views

### Flat Principal Connections and Homotopy Groups?

I've come across a comment in a paper that hints at a relation between the set of (flat) connections for a principal bundle and the homotopy group of the base of the bundle. Can anyone tell me what ...

**4**

votes

**0**answers

412 views

### Flat connections with logarithmic poles and p-adic parallel transport under choice of branch of logarithm

Consider the following simple situation: We work over the ring $R=\mathbb{Z}_p[[t]]$, over which we consider a rank $2$ free module $M$ with basis $(e,f)$. On $M$, we define a flat (topologically ...

**1**

vote

**1**answer

392 views

### notion of connection on Torsors

Hi,
Can anyone please enlighten me on the notion of connection for G-torsors?
Edit (by WW) For some reason the OP decides to add clarification below as an answer rather than editing the question. ...

**3**

votes

**0**answers

552 views

### Gauss--Manin connection for de Rham realisation

Let $X$ and $S$ be smooth schemes of finite type over a field $k$ and let $\pi:X\to S$ be a smooth morphism of finite type. The relative de Rham cohomology $H^i_{dR}(X/S)$ of $X$ over $S$ is a ...

**8**

votes

**1**answer

820 views

### Frobenius Descent

Let $S$ be a scheme of positive characteristic $p$ and $X$ a smooth $S$-scheme. Let $F:X\rightarrow X^{(p)}$ denote the relative Frobenius. A result by Cartier (often called Cartier descent or ...