Questions tagged [derham-cohomology]
Better spelling "DeRham", not derham... I can't figure out how to change this... moderators? The cohomology of the complex of differential forms on a smooth manifold with differential given by exterior derivative.
117 questions
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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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Algebraic de Rham cohomology - open subvariety and normal crossing
I need to do some explicit computation with algebraic de Rham cohomology on some projective varieties, also some open subsets of them.
I don't know the theory well, but I just search some notes to ...
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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 Z_p})...
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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
$$\sum_I ...
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algebraic de Rham cohomology of singular varieties
Hi,
Is there a simple example of an (affine) algebraic variety $X$ over $\mathbb C$ where
the $H^*_{dR}(X/\mathbb C) = H^*(\Omega^\bullet_{A/\mathbb C})$ differs from the singular cohomology $H^*_{...
- 2,842
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Mayer-Vietoris on Fibered Products
Suppose $M \xrightarrow{p} X$ is a surjective submersion of smooth manifolds. Define the fibered product in this case: $$M \times_X M = \{ (m_1, m_2) \in M \times M | p(m_1) = p(m_2) \}$$
and let $U =...
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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 ...
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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 ...
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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$ ...
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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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Relative De Rham cohomologies
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) ...
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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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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
$H^{p+1}(\mathcal{O}_X)=H^{p+1}(\Omega^1_X)=...=H^{p+1}(\Omega^{p-2}...
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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 ...
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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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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 ...
- 2,842
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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
$$ \...
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