Questions tagged [de-rham-cohomology]

This a cohomogy theory for smooth manifolds, where the chain complex is defined by "closed n-forms" / "exact n-forms".

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48 views

algebraic de Rham cohomology of blowup of relative line

Let $$T = \mathbb{A}^1_k,\,\,\, Y = \mathbb{P}^1_T,\,\,\, X = \mathscr{B}(Y),$$ the blowup of $Y$ at a point. I am trying to compute the de Rham cohomology $H^1_{dR}(X/T)$, but I could use some help. ...
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Tu's An Introduction to Manifolds - Section 26.2 Cohomology of a circle, tabular form.

I'm trying to understand how to use the Mayer-Vietoris sequence to compute Cohomologies. There's a small chapter in Tu's Introduction to Manifolds explaining the basics, with some basic theory. More ...
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De Rham cohomology of $S^1$ with compact supports (Bott/Tu)

This is a question about Example 2.9, in Bott/Tu - Differential Forms in Algebraic Topology. Consider the decomposition of $S^1=U\cup V$ by two open sets, as in the figure above. Then both $U$ and $V$...
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First sheaf cohomology of closed 1-forms on a homogeneous space

Given a smooth quasi-projective variety $X$ over $\mathbb C$, is there a good way to compute the first sheaf cohomologiy $H^1(X,\Omega_{X,cl}^1)$ of closed 1-forms $\Omega_{X,cl}^1$? What I'm mostly ...
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Commutative diagram in Poincare dulality ( de Rham cohomology)

Poincaré duality gives us, for a smooth orientable $n$-manifold $M$ with boundary $\partial M$ an isomorphism $I_{M} : H^{k}_{dR}(M)\longrightarrow H_{n-k} (M, \partial M)$ defined as follow $$<J(\...
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Poincaré duality in de Rham cohomology

Poincaré duality gives us, for a smooth orientable $n$-manifold, an isomorphism $$I : H^{k}(M)\longrightarrow H_{n-k}(M), \ \ \gamma\frown [M]$$ between singular homology and cohomology groups where $[...
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Functoriality of twisted de Rham Cohomology

Let $f: M\to N$ be a smooth map of finite-dimensional manifolds, and let $E\to N$ be a flat vector bundle over $N$. Consider the pullback bundle $f^*E\to M$ over $M$ and consider the twisted de Rham-...
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If $X$ is closed then $JX$ is closed

Let $(S, g, J)$ be a closed Riemann surface with a Riemannian metric $g$ compatible with the complex structure $J$. Suppose that a smooth vector field $X$ on $S$ is closed, i.e., the $1$-form $\omega =...
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An action of Lie group and cohomology

Let $X$ be a smooth manifold with an action of a Lie group $G$, i.e. at least we have a map $G \times X \to X$ of smooth manifolds. Then apply the cohomology functor and obtain $H^i (X, \mathbb{R}) \...
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What is a good reference to cite for the definition of forms in terms of germs of functions?

I would like to edit the Wikipedia articles [[Differential (infinitesimal)]] and [[Differential form]] to mention the definition of 1-forms in terms of germs of functions. None of the appropriate ...
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Relation between Hochschild homology and cohomology

Let $A$ be an associative algebra, then we have the Hochschild chain complex, namely: $.. \to A^{\otimes 3} \xrightarrow{d_2} A^{\otimes 2} \xrightarrow{d_1} A$, where, for example, $d_1 (a \otimes b)...
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Example 24.4 Tu's An Introduction to Manifolds

I'm trying to understand the example 24.4 (De Rham Cohomology of a circle), which also contains a lemma. I'll write down the all example and I'll write my questions Let $S^1$ be the unit circle in ...
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Do the Generators of integral de Rham Cohomology always integrate to 1?

Let $\omega$ be the Fubini-Study-Form on complex projective space $\mathbb{P}^N$, normalized s.t. $\int_{\mathbb{P}^1}\omega =1$, and let $1\leq n\leq N$. Then $\int_{\mathbb{P}^n}\omega^n =1$. I've ...
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de Rham cohomology with compact support isomorphic to $\mathbb{R}$

The $k$-dimensional de Rham cohomology with compact support is defined as $$H_c^k(M)=\frac{Z_c^k(M)}{B_c^k(M)} $$ where $Z_c^k(M)$ is the vector space of all closed $k$-forms with compact support on $...
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How is de Rham cohomology useful?

I'm learning about smooth manifolds from the last part of Introduction to Manifolds. Despite the fact I read this part few times I feel like I can't master it because I need to understand how it is ...
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The Hodge diamond of a Calabi-Yau Fourfold

I am studying the String Theory book by Becker, Becker, and Schwarz, and I decided to verify the Hodge diamonds for a CY3 and a CY4. These can be found on page 365 and they are eq.(9.14) and (9.16). ...
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Why is every top form on a non-compact manifold exact? (without invoking Poincare duality)

Let $M$ be a connected, orientable $n$-manifold without boundary. A well known fact is that the top cohomology $H^n(M, \mathbb{R})$ vanishes if and only if $M$ is compact, but I have not been able to ...
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Notation of de Rham Complex of a Manifold

I am confused by some notation used in Loring Tu's An Introduction to Manifolds (2nd edition) to describe the de Rham complex on a manifold. Below I quote the relevant portion from Chapter 25 (page no....
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Geometrical analogue of differential sheaf

If $X$ is an $S-$scheme, then we have a quasi-coherent sheaf $\Omega^1_{X/S}$ on $X$. I'm wondering if there's an analogue for smooth manifolds. For example, let $f:X\to Y$ be a smooth morphism ...
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Normalization of the generator of third cohomology of a compact Lie group

It is proven in "Loop groups" by Pressley and Segal (Prop. 4.4.5, p. 49) that the left invariant 3-form $\sigma$ on a simply-connected compact Lie group $G$ whose value at the identity is given by $$ \...
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Poincare Duality and Integration Isomorphism

Corollary 10.27 in Jeffrey Lee's book "Manifolds and Differential Geometry" states that If $M$ is a connected oriented $n$-manifold with finite good cover, then $H^n_c (M) \simeq \mathbb{R}$. ...
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Exactness of a 1-form on $\Bbb R^3\setminus \{\Bbb S^1_{xy}, (0,0,z)\}$

I want to show that (by calculus) a 1-form $\omega$ on $\Bbb R^3\setminus \{\Bbb S^1_{xy}(0,0,0), (0,0,z)\}$ is exact $\iff$ it is closed and satisfies the following conditions: $$\int_{\Bbb S^1_{xy}(...
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Solution verification for $\int_{M}\omega =0 \iff \omega=d\alpha$

I want to prove that in a compact smooth orientable manifold $M$ (without boundary) of dimension $n$ $$\int_{M}\omega =0 \iff \omega=d\alpha,\quad \omega\in \Omega^n(M), \alpha\in \Omega^{n-1}(M).$$ ...
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Poincaré duality for de Rham cohomology on non-compact manifolds

Let $M$ be an $n$-dimensional orientable non-compact manifold. Is there an isomorphism as follows, and if so how can we construct it? (Or can you provide a reference?) $$ H^{n-i}_{\operatorname{...
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symplectic form, tautological 1-form and DeRham cohomology

There's a question in Lee's Introduction to Smooth Manifold that asks to prove that the Grassmannian product of a symplectic form is not exact. However, isn't this incorrect if there exists a ...
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Differential forms and hairy ball theorem

Take the $2$-form $\omega = \sin\phi \,\mathrm d \phi \wedge \mathrm d \theta$ in spherical coordinates, defined on the portion of $\mathbb R^3$ outside of the unit sphere. Clearly, $\omega$ is closed ...
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63 views

Is there an inverse of the Stokes' Theorem for manifolds with boundary?

I would like to consider this question When does a null integral implies that a form is exact? (also related to Top deRham cohomology group of a compact orientable manifold is 1-dimensional), but for ...
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Cohomology of $T^{2} \times \mathbb{R}^{2}$

I need to find the cohomology groups $H^{k}_{dR}$ of $T^{2} \times \mathbb{R}^{2}$ (where by $T^{2}$ I mean the two-torus). I would like to use the Mayer Vietoris sequence. Any advice on which open ...
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Resolution of the constant sheaf $\mathbb C$.

The differential-graded algebra of complex-valued smooth differential operators $\mathcal{A}_{X, \mathbb C}^{\bullet}$ on an $n$-dimensional complex manifold $X$ (real dimension $=2n$) is acyclic ...
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Doubt about obtaining integral cohomology form from a rational cohomology form

The above text is from Griffiths and Harris. $M=V/\Lambda$ is a complex tori where $V$ is a $n$-dimensional complex vector space and $\Lambda$ is a lattice isomorphic to $\mathbb{Z}^{2n}$ in $V$. We ...
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Computing the compactly supported de-Rham cohomolgy $H^{1}_{c}(\mathbb{R})$

I am asked to show that $H^{1}_{c}(\mathbb{R})\cong\mathbb{R}$. In coordinates, a compactly supported 1-form on $\mathbb{R}$ can be written as $\omega=f(x)\mathrm{d}x$, where $f(x)$ is a compactly ...
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Cohomology of $G$-invariant differential forms

Let $G$ be a Lie group with a left action on a manifold $M$, $\cdot : G \times M \to M$. Define a $G$-invariant differential form $\alpha \in \Omega^{k}(M)$ as a form satisfying $g^{*}\alpha = \alpha,...
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understand Torsion 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 arised a discussion when the map $H^2(X,\mathbb{...
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Fiber of direct image of relative de Rham complex in non-proper case

Let $f: X \rightarrow B$ be a smooth family of complex varieties ($X$ and $B$ are also smooth). If $f$ is proper, then the direct image of the relative de Rham complex, $Rf_*\Omega_{X/B}^{\bullet}$,...
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Kummer suface ; cohomology of the resolution

I have questions regarded to the resolution of Kummer surface. You can see the other 2 ones here At first, I am describing a resolution of Kummer surface: Get a lattice of rank 4 ; $\Gamma$ on $ \...
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Excision for Relative de Rham Cohomology

I recently learned in Bott-Tu about the notion of relative de Rham cohomology, which is defined as follows: If $M$ is a smooth manifold and $S\subset M$ is its (embedded) submanifold, we define ...
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Lefschetz Theorems (Complex Geo by D Huybrechts)

I have a couple of questions on a proof from Daniel Huybrechts' Complex Geometry Complex Geometry on pages $134:$ Corollary 3.3.6 If $X$ is a compact Kahler manifold, then $Pic^°(X)$ is in a ...
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Algebraic de Rham cohomology for projective hypersurfaces degree d

I am trying to understand how to get hold of the algebraic de Rham cohomology for a smooth projective hypersurface (say the zero set of some polynomial $f$). I think I need to find a hypercohomology(?)...
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63 views

De Rham cohomology group of disjoint union

I'm studying de Rham cohomology, but there is something not quite clear. Consider a manifold $M$ and $U_i \subset M$ for $i\in I$ a collection of open sets. I can't figure out why when computing the ...
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Quotient by the image of a zero map

I'm studying a bit of de Rham Cohomology and trying to understand the following step in the proof of a lemma: [...] By definition, $H^{0} = \frac{Ker(d:\Omega^{0}(U) \to \Omega^{1}(U))}{Im(d:\...
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Proving the de Rham cohomology of $M$ and $M \times \mathbb R$ are isomorphic without homotopy

Given a smooth manifold $M$, is there an elementary way of showing $H^k_{dR} (M) \cong H^k_{dR} (M \times \mathbb R)$, i.e. straight from the definitions of cohomology being the quotient of closed ...
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Global existence of solutions to a differential equation

Let $M$ be a (compact) manifold and $\lambda$ a closed 1-form on $M$. Under which condition on $\lambda$ does there exists a non-trivial (complex) function $F$ such that $$dF = F\lambda \quad \quad ?$...
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is there such a thing as a 3-boundary or “hypersurface” in $\mathbb{R}^3$? RE:de Rham chain complexes.

This set of lecture notes asserts that there is such a thing as a hypersurface, or 3-boundary in $\mathbb{R}^3$. To me a boundary is a set of limit points of a set for which every open ball centered ...
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Isomorphism between space of differential forms and space of basic forms on pricipal $S^1$-bundle and independence of cohomology class

Let $M$ be be a principal $S^1$-bundle, i.e. a smooth manifold with a smooth $S^1$-action (multiplication to the right) that has no fixed points such that the orbits through every point $p \in M$, ...
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Find the DeRham cohomology of the spaces minus a line and a circle

Find the DeRham's cohomology of the following open sets, and then looking at the product conclude that they are not diffeomorphic. a) $M=\mathbb{R}^3\setminus (L_1\cup C)$, $L_1=\{x=y=0\}$ and $C=\{...
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Definition of differential forms from derivations

I was skimming through Taylor's book "Several complex variables with connections to algebraic geometry and Lie groups" (first chapter was the best introduction i have ever read!), glimpsed to section ...
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De Rham's cohomology groups by definition

I have to find the De Rham's cohomology groups of $X$, where: a)$X=S^1\times S^1$. b) $X=\mathbb{R}^3\setminus \mathbb{R}$. c) $X=\mathbb{R}^3\setminus S^1$. (d) $X=\mathbb{R}^3\setminus (L_1\cup C)$...
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What controls the central extensions of Lie algebras when the Lie Group is not compact?

If the Lie algebra $\mathfrak{g}$ can be realized as the tangent space of a compact Lie group $G$, then all the possible central extensions of $\mathfrak{g}$ are in one to one correspondence which the ...
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196 views

Definition of the cup (wedge) product of de Rham cohomology classes

In a few places (Lemma 3.0.13 of this script, discussion after Lemma 3.2 here, Proposition 5 here, etc.) I've noticed a certain omission in the definition of a wedge product $\wedge\colon H^k(M) \...
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Bott and Tu Proposition 12.1

Question about the proof of Bott and Tu's Proposition 12.1: Given any double complex $K$, if $H_\delta H_d(K)$ has entries only in one row, then $H_\delta H_d$ is isomorphic to $H_D$. Remark: $\delta$...