All Questions
Tagged with derham-cohomology dg.differential-geometry
31 questions
38
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) ...
- 830
17
votes
1
answer
1k
views
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 $F_A$ ...
- 1,761
13
votes
1
answer
725
views
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 ...
- 1,346
13
votes
1
answer
2k
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 ...
- 303
9
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?...
- 10.5k
9
votes
0
answers
640
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 ...
- 91
9
votes
0
answers
347
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 $i^* \star \omega = 0$ ...
- 91
7
votes
2
answers
429
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 ...
- 2,044
6
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 ...
- 151
6
votes
1
answer
374
views
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 ...
- 6,023
6
votes
0
answers
156
views
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
...
- 6,023
5
votes
2
answers
361
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(\...
- 123
5
votes
2
answers
2k
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 ...
- 3,399
5
votes
1
answer
634
views
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 ...
- 356
5
votes
0
answers
248
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
4
votes
1
answer
211
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. $b_p\geq 1$. ...
- 4,195
4
votes
0
answers
109
views
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^* ...
user126256
3
votes
1
answer
203
views
Cohomology of the complex of differential forms with Schwartz coefficients
Let $U$ be an open manifold (say an open subset of $\mathbb{R}^n$ for simplicity). Denote by $\mathscr{S}(U)$ the space of Schwartz functions on $U$. Schwartz functions are defined as usual to be ...
- 1,374
3
votes
1
answer
167
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 ...
- 6,023
3
votes
0
answers
96
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=\...
- 31
2
votes
1
answer
200
views
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}^...
- 993
2
votes
1
answer
839
views
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 $\mathcal{X}$.
...
- 6,023
2
votes
0
answers
241
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 ...
- 2,907
2
votes
0
answers
130
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 ...
2
votes
0
answers
152
views
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^...
- 5,891
2
votes
0
answers
327
views
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 \...
- 1,379
2
votes
0
answers
190
views
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 $\...
- 245
1
vote
1
answer
239
views
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 $L_X \alpha =0$?
- 356
1
vote
1
answer
788
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 ...
- 23
1
vote
0
answers
82
views
Projection to trivial reduced cohomology class in $L^2(\mathbb{R})$
Given that I have had no success on the mathematics stackexchange (see here), I've decided to try my luck here.
I am attempting to solve the following exercise (original formulation here), which to my ...
- 111
0
votes
0
answers
85
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}...
