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0
votes
0answers
72 views

Comparison of support of a divisor and its class in de Rham cohomology

Let $X$ be a smooth surface over $\mathbb{C}$ and $D$ be an effective divisor. Suppose that the linear system corresponding to $D$ is zero dimensional. Denote by $[D] \in H^2(X,\mathbb{Z})$ the ...
1
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0answers
211 views

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 ...
6
votes
3answers
476 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 ...
1
vote
1answer
159 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 $k = \mathbb{C}$.) ...
2
votes
2answers
203 views

De Rham isomorphism for noncompact manifolds?

Maybe someone has a quick answer. Thanks. For noncompact manifolds, is the De Rham cohomology isomorphic to the singular cohomology? Is the De Rham cohomology defined with the cochain of compactly ...
5
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0answers
208 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 ...
5
votes
1answer
257 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 ...
3
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0answers
139 views

Hodge filtration over $\mathbb Z_p$

Let $p$ be a prime number. Let $X\to\operatorname{Spec}\mathbb Z_p$ be smooth and proper. Is it true that the map $H^i(X,\Omega^{\bullet\geq j}_{X/\mathbb Z_p})\to H^i(X,\Omega^\bullet_{X/\mathbb ...
5
votes
2answers
566 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 ...
4
votes
1answer
299 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 ...
1
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0answers
144 views

Mayer-Vietoris on Fibered Products

Suppose $M \xrightarrow{p} X$ is a surjective submersion of smooth manifolds. Define the fibered product in this case: $$M \times_X M = \{ (m_1, m_2) \in M \times M | p(m_1) = p(m_2) \}$$ and let $U ...
1
vote
1answer
241 views

deRham cohomology of a manifold with covering space $S^{n}$

(A qual problem) Let $\pi:S^{n}\rightarrow M$ be a covering map, $M$ being an orientable manifold. Show that $H_{deR}^{k}(M)=0$ for $1\leq k < n $. We can show $H_{deR}^{1}(M)=0$ by the following ...
4
votes
1answer
466 views

de rham model for relative cohomology

In GTM82, I read a model for the relative cohomology of (M,N) with N a submanifold of M. And in the page: Relative De Rham cohomologies , I got to know that there is another model for relative ...
9
votes
0answers
170 views

Is there a Hodge isomorphism theorem for part-tangential, part-normal, harmonic differential forms?

Let $M$ be an oriented compact Riemannian $n$-manifold with boundary $\partial M$. A differential $p$-form $\omega$ on $M$ is normal if $i^* \omega = 0$ holds, tangential if $i^* \star \omega = 0$ ...
5
votes
1answer
383 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 = ...
9
votes
1answer
672 views

Relative De Rham cohomologies

Hello, as far as I know, there are two main ways to have a relative version of De Rham Cohomology for a pair (M,N), where M and N are smooth manifolds and N is a closed (as a topological subspace) ...
10
votes
1answer
528 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 ...
1
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0answers
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 ...
2
votes
2answers
459 views

Global Definition of the Dolbeault Complex of a Vector Bundle

For an $2n$-dimensional complex manifold $M$, and a smooth vector bundle $E$ over $M$, it is well-known (see Voisin, Huybrechts) that there exists an operator $\overline{\partial}$, built locally from ...
3
votes
0answers
362 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 ...
2
votes
1answer
369 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 ...
2
votes
1answer
395 views

Homology of a region of the plane

This is related to this MO question, I don't know if it's really "research-level". As in that question, let $U$ be a domain of the complex plane $\mathbb{C}$, i.e. an open connected subset. Let $$ ...