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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Different definitions for integral de Rham cohomology classes
Suppose that $S$ is a compact orientable surface. In this case, the top de Rham cohomology space $H^2(S)\cong \mathbb{R}$, with the isomorphism given by integration on $2$-forms along $S$.
Now, one ...
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The unit root subspace of a genus-2 odd degree hyperelliptic curve of semistable reduction
Let $K$ be a finite extension of $\mathbb{Q}_p$. If $A_K$ is a semistable Abelian variety over $K$, then we have a Frobenius endomorphism on $H_{dR}^1(A_K)$, whose definition depends on a choice of a ...
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how to see the Gysin map explicitly in an easy situation
Let $C$ be a smooth projective curve and let $U \subset C$ be an open affine subset, with closed complement $S$ consisting of a finite number of points. I am trying to see explicitly the Gysin map in ...
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algebraic de Rham cohomology of toric varieties (reference request)
I haven't been able to find anything workable yet, but I'm looking for a reference on the de Rham cohomology of toric varieties, where as many as possible of the following conditions are handled:
...
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Degeneration twisted Hodge to de Rham spectral sequence
Let $X$ be a proper and smooth scheme over $\mathbf{C}$ and let $\mathbb{L}$ be a local system of finite dimensional $\mathbf{C}$-vector spaces. By the Riemann Hilbert correspondence, to $\mathbb{L}$ ...
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How does one compute the space of algebraic global differential forms $\Omega^i(X)$ on an affine complex scheme $X$?
In 1963 Grothendieck introduced the algebraic de Rham cohomolog in a letter to Atiyah, later published in the Publications Mathématiques de l'IHES, N°29.
If $X$ is an algebraic scheme over $\mathbb C$...
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Understanding the Exercise 9.9.5 of Weibel homological algebra
The exercise 9.9.5 of Weibel's homological algebra states that
$\textbf{Exercises 9.9.5}$ If $Z$ is a nilpotent ideal of $R$ and $k$ is a field of $char(k) = 0$, show that $H_{dR}^{\ast}(R) \cong H_{...
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Is the action of the absolute Frobenius on de Rham cohomology induced by an algebraic map?
Let $X\to \mathrm{Spec}\:\mathbb{Z}_{(p)}$ be a smooth proper morphism with a geometrically connected generic fiber. Assume that the special fiber has an $\mathbb{F}_p$-point.
Via the isomorphism $H^{*...
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De Rham's theorem for top-forms in manifolds with boundary
In page 79 of Bott-Tu, "Differential Forms in Algebraic Topology", they define the relative de Rham theory as follows:
Let $f:S\to M$ be a smooth map. Define the complex $\Omega^*(f)$ by
$$\...
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Can Hodge symmetry fail if there is a lift to $W_2$ and the crystalline cohomology is torsion-free?
Let $f:X\to \mathrm{Spec}\:\mathbb{F}_p$ be a smooth proper morphism with $p>\mathrm{dim}\:X$. Assume that $H^i_{\mathrm{crys}}(X/\mathbb{Z}_p)$ is torsion-free for all $i\geq 0$ and that there is ...
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Sheaf / de Rham cohomology of a stack with values in a complex of abelian sheaves
I am reading Differentiable Stacks and Gerbes to understand about (hyper) cohomology groups of a stack $\mathcal{X}$ with values in a complex $\mathcal{M}$ of abelian sheaves over $\mathcal{X}$.
...
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When are the cotangent and tangent sheaves isomorphic?
Let $X$ be an $S$-scheme. Under what conditions, if any, is the cotangent sheaf $\Omega_{X/S}$ isomorphic to the tangent sheaf $\Theta_{X/S}$ as $\mathcal{O}_X$- modules? For example, given a ...
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Geometric theory for cohomology groups $H^p(M;\mathbb{Z})$
An excerpt from the book Loop Spaces, Characteristic Classes and Geometric Quantization by Jean-Luc Brylinski is mentioned below:
Characteristic classes are certain cohomology classes associated
...
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On a mysterious reference of Grothendieck
These days I found a mysterious page on Google books describing a book entitled On the De Rham cohomology of schemes by Grothendieck, Coates, and Jussila.
At once I thought this was an error and ...
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Relative de Rham Cohomology groups of k-algebra
Let $A$ be a commutative unital $k$-algebra. Then we have de Rham complex given as:
$C_{\ast}(A)$ : $ 0 \rightarrow A \rightarrow \Omega_{A \lvert k}^{1} \rightarrow \Omega_{A \lvert k}^{2} \...
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Find torsion classes using flat bundles
My question refers to a discussion from this older thread on Neron-Severi group of a Kähler manifold. In the comments below Ted Shifrin's answer there arose a discussion when the map $H^2(X,\mathbb{Z}...
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Direct proof that Chern-Weil theory yields integral classes
Suppose $E$ is a complex vector bundle of rank $n$ on a compact oriented manifold (both assumed smooth). Let $h$ be a Hermitian metric on $E$, and let $A$ be a Hermitian connection on $E$ and $F_A$ ...
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Multidifferential operators with vanishing integrals
(Moved from math.stackexchange.)
Is the following proposition true?
Given a multidifferential operator $D$ on $\mathbb{R}^n$ with constant coefficients, i.e. for all functions $f_1,\dots,f_k \in C^...
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A variant on characteristic $p$ de Rham cohomology
I was thinking about de Rham cohomology in characteristic $p$, and in particular the recent question about Poincare residues, and I came up with the following construction.
Let $k$ be a perfect field ...
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When are automorphisms of the cohomology ring realized by isometries?
Let $(M,g)$ be a closed smooth Riemannian manifold, and denote by $G$ a closed subgroup of its isometry group. By considering the maps $g^*$ induced by elements $g\in G$ in the (de Rham) cohomology $H^...
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Easier ways to compute homology/cohomology by adding extra structure
Suppose $X$ is a topological space and I want to talk about its “homology”.
There is this notion of singular homology obtained from singular chain complex. This is not very easy to compute.
Suppose ...
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Generalized de Rham cohomology on product bundle giving specified cohomology
Given a compact, smooth manifold $M$ and a real vector bundle $E \to M$ (in general not flat). There already have been numerous questions about how to equip the space $\bigoplus_k \Gamma(\Lambda^k T^* ...
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vanishing of higher algebraic de Rham cohomology and sheaves of differentials for singular curves
I'm looking for some results or references about de Rham cohomology of curves in less-than-optimal cases. The two vanishing results I care about are:
$H^{k}_{\text{dR}}(X) = 0$ for $k>2$, since $H^...
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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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De Rham cohomology of Lie groupoid
Let $G$ be a Lie group acting on a manifold $M$.
Consider the transformation groupoid $\mathcal{G}=(G\times M\rightrightarrows M)$. We have the notion of de Rham cohomology of a Lie groupoid by ...
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Suppose that two cohomologous forms agree on every restriction. Do they agree?
Let $\eta$, $\omega$ be two $(1,1)$-forms on $\mathbb{C}^m \times Y$, where $Y$ is a compact Kahler manifold with vanishing first Chern class, i.e., a Calabi-Yau manifold. Suppose that for all $z \in \...
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Leafwise de Rham cohomology (A true definition of differential forms along leaves)
For a foliated space $(M, \mathcal{F})$, one associate a leafwise de Rham cohomology. This cohomology and trace-class operators on this cohomology and trace interpretations for closed orbits of ...
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De Rham cohomology and extension of scalars
Let $K$ be a field of characteristic zero and let $X$ be a smooth variety over $K$. Given a field extension $L/K$, I denote by $X_L$ the variety $X \times_{Spec(K)} Spec(L)$.
What is the easiest way ...
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Vanishing product of a closed and coclosed form on a Riemannian manifold
For a (compact) Riemannian manifold $(M,g)$, can it happen that for a non-zero form $\text{d}^*\omega$, and a smooth function $f$ such that $\text{d}f \neq 0$, we can have
$$
\text{d}f \wedge \text{d}^...
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Alternating property of H_2(T, Z)
Let us consider the torus $T = S^1_X \times S^1_Y$, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. We have an alternate property $dX \wedge dY = - dY \wedge dX \...
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Formality of surfaces
The de Rham dg algebra $\Omega(F)$
of a closed orientable surface $F$
is formal
(that is, weakly equivalent to its cohomology algebra).
This is a special case of the fact of formality of Kähler ...
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Classification of line bundles by second cohomology of a manifold
In the book Loop spaces, Characteristic classes and geometric quantization by Brylinski I see following result when trying to motivate geometric description of $H^3(M,\mathbb{Z})$.
$H^2(M,\mathbb{Z}...
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Definition of algebraic de Rham cohomology of non-smooth affine variety
$\newcommand{\Hdr}{H_{\mathrm{dRh}}}$
$\newcommand{\spec}[1]{\mathrm{spec}(#1)}$
$\require{amsmath}$
Let $A = k[x_1,\ldots,x_n]$ the polynomial ring over a field $k$ of characteristic zero and $I \...
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Can every De Rham cohomology class be represented by a closed form $\alpha$ with $L_X \alpha=0$
Assume that $M$ is a manifold and $X$ is a vector field on $M$.
Is it true to say that every closed form is De Rham-cohomologue to a closed form $\alpha$ with $L_X \alpha =0$?
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Comparison for cycle class maps for de Rham and etale cohomology via p-adic Hodge theory
Let $K$ be a p-adic local field, $X$ a smooth projective variety over $Spec(K)$, $CH^k(X)$ the Chow group of pure codimension $k$. Then there are cycle maps
$cl^X_{DR}:CH^k(X)\to H^{2k}_{DR}(X/K)$ ...
CommunityBot
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Which compact (finite dimensional) Lie groups have $H^1_{DR}(G)\neq 0$
In particular, I am wondering if $H^1_{DR}(G)\neq 0$ implies that the group can written as a semidirect product of $\mathbb{S^1}$ and something else, with the $\mathbb{S^1}$ factor being responsible ...
CommunityBot
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What is the scope of validity of Kunneth formula for de Rham?
In books like Bott-Tu or all pdf texts I have found on internet, the Kunneth formula for manifolds $M$ and $N$ and their de Rham cohomology
$$ H^{\bullet}_{dR}(M \times N) \simeq H^{\bullet}_{dR}(M) \...
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Counterexample showing that G-invariant de Rham cohomology different from cohomology of G-invariant sub-complex?
If $G$ is a discrete or a Lie Group acting smoothly on a manifold $M$, we can define the algebra of $G$-invariant de Rham classes, $H(M)^G$, and we can also consider the cohomology of the sub-complex ...
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How does one introduce characteristic classes [closed]
How does one introduce, or how were you introduced to characteristic classes?
You can assume that the student is comfortable with principal bundles and connections on principal bundles.
I am not ...
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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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Question on de Rham complex with distributional coefficients
Let $X$ be a smooth manifold (usually assumed to be paracompact). Let us denote by $\underline{\Omega}^{p,-\infty}_X$ the sheaf of real valued $p$-forms with distributional coefficients in the ...
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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^*_{...
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Lagrangian up to Hamiltonian in cotangent bundle
I want to understand the folklore conjecture that, in a CY manifold, Lagrangians up to Hamiltonian isotopies are represented by special Lagrangians by examining cotangent bundle and Hodge theory.
...
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Countability assumption for good covers in Bott-Tu
In chapter II of their text Differential Forms in Algebraic Topology, Bott and Tu construct the Čech-De Rham complex with regards to an open covering indexed by some ordered and countable indexing set....
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What is known about this type of generalisation of de Rham cohomology?
I will describe a certain generalisation of de Rham cohomology; things could be generalised further but I will stick to a concrete example. A $0$-double-form is a function on the complex plane $\...
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naive de Rham cohomology fails for singular varieties
Let $X$ be a variety over a field $k$ of characteristic zero. If $X$ is smooth, algebraic de Rham cohomology defined as
$$
H^n_{dR}(X / k)=\mathbb{H}^n(X, \Omega^\bullet_{X/k})\qquad (\star)
$$ is a ...
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Cohomology classes functorial under etale morphisms
Consider the full subcategory $\mathcal C$ in the big etale site of a field $k$ consisting of smooth schemes, namely the category of smooth varieties over a field $k$ with etale maps as morphisms.
I ...
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Proof of Lefschetz-Hopf Fixpoint Theorem with de Rham cohomology?
Looking for a proof of the Lefschetz-Hopf Fixpoint Theorem with the de Rham Cohomology.
(I´m more interestet in the Formula then just the simple statement that if the Lefschetz number is not zero ...
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de rham model for relative cohomology
In GTM82, I read a model for the relative cohomology of (M,N) with N a submanifold of M.
And in the page:
Relative De Rham cohomologies,
I got to know that there is another model for relative ...
CommunityBot
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Reference: Relative cohomology of a morphism
Let $f\colon Y \to X$ be a morphism of schemes, the inverse image in $K$-theory always fit into a long exact sequence
$$
\cdots \to K_i(f)\to K_i(X) \xrightarrow {f^*} K_i(Y)\to \cdots
$$
where the ...
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