Questions tagged [derham-cohomology]
The cohomology of the complex of differential forms on a smooth manifold with differential given by exterior derivative.
107
questions
4
votes
0
answers
122
views
Nice proof that de Rham complex computes Lie algebra cohomology?
If $G$ is a nice enough group acting on a nice enough space $X$, then the relative de Rham complex
$$\Omega^\bullet_{X/(X/G)}\ \simeq\ \mathcal{O}_X\otimes\text{Sym}\,\mathfrak{g}^*[-1]$$
is given by (...
- 5,565
0
votes
0
answers
71
views
Existence of covering space with trivial pullback map on $H^1$
I have seen somewhere the following claim (but can't remember where): let $M$ be a connected orientable closed smooth manifold with $b_1(M)=1$, then there exists a connected covering space $p:\tilde{M}...
2
votes
1
answer
464
views
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 ...
- 4,618
1
vote
0
answers
137
views
Calculation of de Rham cohomology of abelian varieties/ jacobian varieties
It's known that for a elliptic curve like $E:y^2=x(x-1)(x-t)$ we have a basis $\frac{dx}{y}, \frac{xdx}{y}$ for $H_{dR}^1(E)$. But find such a basis is not an easy thing. I wonder for a general ...
- 523
3
votes
1
answer
247
views
Calculation of Frobenius on de Rham cohomology of elliptic curves with good reduction
I'm reading "An introduction to the theory of $p$-adic representations" by Laurent Berger. In the page 14 it says:
Suppose $F$ is an unramified finite extension of $\mathbb Q_p$ and $E$ is ...
- 36.2k
5
votes
0
answers
219
views
Algebraic de Rham cohomology with torus coefficients
Let $X$ be a smooth projective variety over $\mathbb{C}.$
On page 3 in this preprint of Simpson, it is stated that
Notice first of all that the algebraic de Rham theory is not going to work well in ...
- 123
5
votes
2
answers
347
views
Exterior differentiation of foliations
Let $M$ be a differentiable manifold.
Let $T^*M$ be the cotangent bundle of $M$.
Consider the exterior differentiation $d: A^p(M)\longrightarrow A^{p+1}(M)$, where $A^p(M)=\Gamma(\...
- 2,329
5
votes
1
answer
202
views
Gysin isomorphism in de Rham cohomology using currents
I'd like to find a reference for the following fact.
First, some background: we can define de Rham cohomology of a smooth manifold $X$ of dimension $d$ using the de Rham complex
$$
\Omega^0_X\to \...
- 6,666
7
votes
1
answer
374
views
Is anything known about de Rham $K(\pi,1)$'s?
Let $X$ be a connected qcqs scheme. We say that $X$ is a (étale) $K(\pi,1)$ if for every locally constant constructible abelian sheaf $\mathscr{F}$ on $X$ and every geometric point $\overline{x}$ the ...
- 15.3k
7
votes
2
answers
404
views
Can one make sense of de Rham cohomology for the complement of a (dense) irrational flow on the torus?
Recent work has led me to consider whether one could define consider the complement of a dense irrational flow on the torus $P_\alpha \subset T^2$ as some kind of generalized smooth space, and ...
- 2,259
4
votes
1
answer
464
views
Clarification on smooth de Rham theorem
I am misunderstanding something in Theorem 2.1.9 in Dimca’s Sheaves in Topology:
Let $X$ be a real smooth manifold. Then the natural morphism from the constant sheaf to the de Rham complex
$$\mathbb{R}...
- 36.2k
5
votes
0
answers
288
views
Generalization of de Rham cohomology, or cohomology for non-smooth case
Let $\Omega\subseteq \mathbb{R}^{3}$ be a contractible region and $f:\Omega\to \mathbb{R}^{3}$ be a vector field on $f$. If $f$ is smooth vector field and $\nabla\cdot f=0$, then $f=\nabla\times g$ ...
- 1,911
2
votes
0
answers
211
views
Monodromy group action on de Rham cohomology
Let $f : Y \longrightarrow X := \mathbb{P}^1\setminus\{0,1,\infty\}$ be the smooth proper morphism associated to the Legendre family, which is an elliptic fibration of the punctured line, with fibre ...
- 1,651
4
votes
0
answers
132
views
Spencer complex and de Rham Complex
in those lectures notes written by Claude Sabbbah: https://perso.pages.math.cnrs.fr/users/claude.sabbah/livres/sabbah_nankai110705.pdf
there is the proposition 1.4.4 where he says that there is a ...
- 385
4
votes
1
answer
202
views
Compute de Rham-Witt sheaves
I am really new to this, but I am having a hard time understanding all the de Rham-Witt construction.
It seems to be really difficult to compute anything with those beasts: like I cannot find any ...
18
votes
0
answers
2k
views
Cycles in algebraic de Rham cohomology
Let $F$ be a number field, $S$ a finite set of places, and $X$ a smooth projective $\mathscr{O}_{F,S}$-scheme with geometrically connected fibers. For each point $t\in \text{Spec}(\mathscr{O}_{F,S})$, ...
- 22.2k
2
votes
0
answers
230
views
Künneth formula for algebraic de Rham cohomology
Let $X$ and $Y$ be finite type schemes over a field $k$, and let $H^i(X/k)$ denote the $i$-th algebraic de Rham cohomology group of $X$ over $k$. I'm interested in the extent to which a Künneth ...
- 60.4k
3
votes
0
answers
125
views
Isomorphism between Weyl and Cartan models as Hom-Tensor Adjunction
Let $M$ be a manifold, $\Omega$ be the de Rham complex of $M$. Let $G$ is a compact Lie group acting on $M$, $\mathfrak g$ its Lie algebra and $W(\mathfrak g) = \Lambda(\mathfrak g^*) \otimes S(\...
- 131
2
votes
0
answers
174
views
Modular forms and Petersson inner product via De Rham cohomology, Hodge filtration and cup products
I'm looking for an explanation on how and why you can define modular forms through De Rham cohomology via the Hodge filtration and especially how the Petersson inner product is related to the cup ...
- 632
3
votes
1
answer
157
views
Models for computing cohomology of Lie groupoids
Given a Lie groupoid $\mathcal{G}=[\mathcal{G}_1\rightrightarrows \mathcal{G}_0]$, let $\mathcal{G}_\bullet$ be the associated simplicial manifold.
Let $\Omega^\bullet(\mathcal{G}_\bullet)$ be the ...
CommunityBot
- 1
16
votes
4
answers
1k
views
Can one glue De Rham cohomology classes on a differential manifolds?
Let $M$ be a differential manifold and $\mathcal H^k$ the presheaf of real vector spaces associating to the open subset $U\subset M$ the $k$-th de Rham cohomology vector space: $\mathcal H^k(U)=H^k_{...
- 46.9k
2
votes
0
answers
110
views
Interpreting the Higher-order Hodge-Laplace Operator
As an operator on functions, one intuitive way to think about the Laplacian seems to be as an operator that returns the average difference between a function's value at a point and the values of its ...
3
votes
0
answers
111
views
Logarithmic differentials on an arithmetic surface, and Poincaré residue
Suppose that $X$ is an arithmetic surface, i.e. $\pi: X \to S$ flat and relative dimension 1 over a Dedekind scheme $S$, and assume $X$ smooth.
Let $Y \subset X$ be a horizontal effective Cartier ...
- 1,338
2
votes
0
answers
156
views
Exact sequence for low-degree terms of relative de Rham cohomology
Let $\varphi:X\to Y$ be a morphism of schemes, smooth of relative dimension 1. The de Rham cohomology $H^\bullet(X/Y)$ is the result of applying the derived functor $R^\bullet\varphi_*$ to the de Rham ...
- 1,338
17
votes
1
answer
640
views
De Rham via topoi
Étale cohomology of schemes $X$ is constructed as follows: one associates to $X$ the so-called étale topos of $X$, and then one just takes the sheaf cohomology of that topos.
Is it possible to ...
- 36.3k
35
votes
4
answers
8k
views
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) ...
- 44k
8
votes
2
answers
1k
views
Hodge dual of de Rham cohomology and singular cohomology
We know that the de Rham cohomology is isomorphic to the singular cohomology, does the Hodge dual of differential forms induce a dual operation on de Rham cohomology, hence also on singular cohomology?...
- 8,726
3
votes
0
answers
94
views
L^1 gradient bounds for potentials of weakly closed forms
Context: The Poincaré-lemma is a central statement in differential geometry. It shows that a k-form is closed iff it is exact. A special case is as follows:
Let $\omega\in\Omega^k(U)$ with
$\omega=\...
- 60.4k
9
votes
1
answer
352
views
Integrating hypercohomology classes
Let $X$ be a complex variety. By Poincare's lemma, its singular cohomology can be computed as hypercohomology of the holomorphic de Rham complex (viewing $X$ as a complex manifold)
$$\text{H}^\cdot(X,\...
- 8,944
14
votes
1
answer
1k
views
Artin vanishing for Stein manifolds and restriction maps
In the setting of complex Stein manifolds $X$ of complex dimension $d$, the theorem of Andreotti--Frankel implies the vanishing of the singular cohomology group $H^i(X,\mathbb Z)=0$ for $i>d$. With ...
- 4,111
2
votes
0
answers
159
views
de Rham cohomology of a specific ring
I'm running into a certain algebraic de Rham cohomology computation I could use some help with. Specifically, what is the algebraic de Rham cohomology of:
$$
\mathbb{C}[x_1,\dots,x_n,y_1,\dots,y_n,(r^...
- 60.4k
9
votes
0
answers
600
views
Does Stokes theorem have anything to do with adjoint functors?
I notice some similarity between Stokes theorem in differential geometry and the definition of adjoint functors:
in both cases, there is a 2-placed function (the $\operatorname{hom}$ functor, or the ...
- 11.4k
4
votes
0
answers
234
views
de Rham Bloch-Ogus theory in positive characteristic
In their famous paper Gersten's conjecture and the homology of schemes, one of the results that Bloch and Ogus prove is that the second page of the coniveau spectral sequence for $X$ smooth over a ...
- 1,964
4
votes
1
answer
532
views
Relationship between Dolbeault and de Rham cohomology on Riemann surface
A lecturer of mine once ``proved'' the existence of non-constant meromorphic functions on a compact Riemann surface $X$ by using analysis of the Laplacian to decompose the de Rham cohomology group as $...
- 34.2k
5
votes
1
answer
2k
views
Integration currents vs Poincaré dual
Let $M$ be a complex manifold of dimension $n$ and $S \subset M$ a closed complex submanifold of complex codimension $r$. Let $[S] \in H_{2r}(S)$ be the fundamental class of $S$.
We have the ...
- 60.4k
2
votes
1
answer
276
views
Where does this clever choice of differential come from? (calculating $\mathrm{H}^1_{\mathrm{dR}}(E/k))$
In these notes of Kedlaya, he calculates the de Rham cohomology of an affine part $X$ of an elliptic curve $E$ over a field $K$, given by $y^2 = P(x) = x^3 + ax + b$.
He uses these relations:
$0 = y^...
- 8,829
6
votes
2
answers
482
views
Künneth formula for de Rham cohomology with respect to an integrable connection
I am reading through https://stacks.math.columbia.edu/tag/0FM9 which proves that for $X,Y$ schemes over some base $S$ and $X \times _S Y \overset{p}{\rightarrow} X$ resp. $X \times _S Y \overset{q}{\...
- 1,069
16
votes
0
answers
816
views
"Geometric" proof of Kunneth formula
The usual proof of the Kunneth formula (say for either the homology or cohomology of manifolds) is essentially pure homological algebra. I was wondering if there was a more geometric proof, i.e., one ...
- 950
1
vote
0
answers
263
views
Berthelot-Ogus comparison isomorphism
On the link, page, $ 2 $, the Berthlot-Ogus isomorphism theorem is stated as follows,
We have a canonical isomorphism, $$ \rho_{\mathrm{cris}} \ : \ H_{\mathrm{cris}}^{i} (X) \otimes_{K_ {0}} K \to H_{...
- 497
4
votes
1
answer
193
views
Existence non-trivial parallel $p$-form implies non-triviality of $p$-th cohomology group using De Rham cohomology
Cross-post from MSE.
Suppose $(M,g)$ be a closed Riemannian manifold. Because every parallel (nontrivial) $p$-form $\omega$ is harmonic so the $p$-th Betti number should be positive i.e. $b_p\geq 1$. ...
- 106k
12
votes
2
answers
1k
views
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 ...
- 151k
4
votes
0
answers
84
views
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 ...
- 41
10
votes
1
answer
380
views
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 ...
- 39.3k
5
votes
0
answers
137
views
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:
...
- 1,102
8
votes
2
answers
426
views
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}$ ...
- 34.2k
17
votes
3
answers
1k
views
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$...
- 3,592
3
votes
0
answers
164
views
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_{...
- 609
4
votes
1
answer
433
views
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^{*...
- 4,064
6
votes
1
answer
771
views
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
$$\...
- 101
4
votes
1
answer
390
views
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 ...
- 7,027