# Tagged Questions

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### Gauss Manin connection in algebraic geometry and DG setting

E.Geltzler defined Gauss Manin connection on periodic cyclic homology of smooth proper DG algebra.I just began to learn his definition.But it seems that his construction is very different to the Gauss ...

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**1**answer

254 views

### Steenrod operations in algebraic geometry

What are some applications of Steenrod operations (or similar constructions) in algebraic geometry?
I am dimly aware of the the use of these Voevodsky's work on motivic cohomology, and would be ...

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**1**answer

122 views

### Most general “finiteness of de Rham cohomology” statement for holonomic $D$-modules in the algebraic case?

Let $X$ be a nonsingular algebraic variety over a field $k$ of characteristic zero. (We may assume $k$ algebraically closed if need be, but I want to avoid specifically demanding $k = \mathbb{C}$.) ...

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164 views

### Hard Lefschetz in De Rham cohomology

I'm looking for a reference for Hard Lefschetz theorem in algebraic De Rham cohomology. By this I mean the statement that
If $i: Y \hookrightarrow X$ is a smooth hyperplane section of a smooth ...

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**1**answer

202 views

### algebraic de rham cohomology of a curve

Let $X$ be a smooth projective curve over a field $k$ of characteristic zero. The algebraic de Rham cohomology of $X$ is, by definition, the hypercohomology of the complex of Kähler differentials for ...

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**2**answers

562 views

### If a $d \log$ form is exact, is it zero?

Let $T = \mathrm{Spec}\ \mathbb{C}[x_1^{\pm 1}, x_2^{\pm 1}, \ldots, x_n^{\pm 1}]$ be an algebraic torus and $X$ a closed subvariety. Let $\eta$ be a differential form on $T$ of the form
$$\sum_I ...

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**1**answer

240 views

### algebraic de Rham cohomology of singular varieties

Hi,
Is there a simple example of an (affine) algebraic variety $X$ over $\mathbb C$ where
the $H^*_{dR}(X/\mathbb C) = H^*(\Omega^\bullet_{A/\mathbb C})$ differs from the singular cohomology ...

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**1**answer

374 views

### Remove denominators in de Rham cohomology

Let $\omega = \mathrm d \eta$ be an exact rational $n$-form on $\Bbb P^n$.
It may happen that the polar locus of $\eta$ is not included in the polar locus of $\omega$. But is it true that $\omega = ...

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**1**answer

507 views

### Monsky's proof of the finiteness of de Rham cohomology

I'd really like to understand the proof that Paul Monsky wrote about the finiteness of the de Rham cohomology of algebraic varieties. I'd like it very much because it seems to explain in concrete ...

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317 views

### Vanishing of cohomology groups

Given a smooth hypersurface $X \subset \mathbb{P}^{2p+1}_{\mathbb{C}}$ of degree $d \ge 5$ (with $p\ge 2$), when can we say that
...

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354 views

### Finite Field Varieties and the de Rham Complex of Kähler Differentials

In an answer to a previous question I asked, where the Kahler differentials of a variety over a finite field were discussed, it was stated that:
You can certainly define de Rham cohomology using ...

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**1**answer

344 views

### algebraic de Rham cohomology functoriality

Suppose that $Y/k$ is a an algebraic variety over a field $k$ of characteristic zero and
that $Y\subseteq X$ is a closed embedding into a smooth variety over $k$. Then the completion
of the de Rham ...