# Tagged Questions

**1**

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**2**answers

337 views

### Quantization by cohomology

Ok, so I have heard some cool stuff here and there about how to Quantize Yang-Mills via cohomology, can anyone refer any texts in the literature that have shed some light on this, I mean I have some ...

**0**

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**0**answers

137 views

### Metalinear frame bundle on sphere or $\mathbb{C}P^n$

Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map ...

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93 views

### Dimension of the set of polarized sections

Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian, i.e. Maximal isotroic, dim$P_m=n$, ...

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**2**answers

297 views

### Isomorphism of connections on a complex line bundle

Reading an article I faced with the following theorem, please give me a reference to a proof of the fact which is stated without any reference in the article. Is it a well-known fact?
Theorem. Let $E ...

**5**

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**1**answer

334 views

### Are the Kahler Identities for a Holomorphic Vector Bundle Actually Interesting?

For a Kahler manifold $M$, and a smooth vector bundle $E$ over $M$, let us denote by $A^{(p,q)} := \Omega^{(p,q)} \otimes E$ the bundle of forms with values in $E$. Now with respect to a choice of ...

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**0**answers

470 views

### Cech Cohomology of Sections of Holomorphic Bundle over Contractible Space

Is there anything that can be said in general about the Cech cohomology of the sheaf $\mathcal{F}$ of sections of a holomorphic bundle over a contractible space $X$?
I know that such a bundle must ...

**6**

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**0**answers

125 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 $i^* \star \omega = 0$ ...

**4**

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**3**answers

333 views

### Connection Transformation Formula; Degree 3 Cech Cohomology

While reading through Brylinski, as in all of my posts, I am trying to understand the following equation:
$ g_* \tilde{\theta} = \tilde{\theta} - g^{-1} dg$
Setting
I have a principal ...

**8**

votes

**1**answer

688 views

### Differential Calculus and the De Rham Homotopy Operator

Suppose $M$ is a smooth manifold, $\Omega^k(M)$ the vector space (or $C^\infty(M)$-module in case this is better suited) of $k$-differentialforms on $M$ and
$$I: \Omega^k(M\times \mathbb{R}) \to ...

**2**

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**2**answers

413 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 ...

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**2**answers

891 views

### How to Draw Complex Line Bundles

I am giving a presentation soon on the Classification of Complex Line Bundles and I would like to have some very "basic" visualizations to use as examples.
Background and Context
I am considering ...

**15**

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**5**answers

1k views

### de Rham vs Dolbeault Cohomology

For a complex manifold $M$, one can consider (A) its de Rham cohomology, or (B) its Dolbeault cohomology. I'm looking for some motivation as to why one would bother introducing Dolbeault cohomology. ...

**5**

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**1**answer

286 views

### Continuous and smooth Lie groupoid cohomology

In the paper by Weinstein and Xu: Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), there are two versions of Lie groupoid cohomology. The same differential ...

**2**

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**1**answer

551 views

### de Rham cohomology vs. iterated tangent bundles?

I have two related questions. Here $M$ is a real smooth manifold, $TM$ is its tangent bundle, $T^n M := T ... TM$ is the $n$-th iterated tangent bundle.
Fiberwise linear smooth functions $TM \to ...

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**1**answer

301 views

### When does the Anti-Holomorphic Chain Complex Exist for Non-Kahler Manifolds?

Given an $N$-dimensional Riemannian manifold $M$, with associated Hodge $\ast$-mapping $\ast$, we have the chain complex
$$
\Omega^{0} {\buildrel {\text d}^\ast \over \longleftarrow} \Omega^{N} ...

**1**

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**1**answer

857 views

### Betti Numbers (homology vs cohomology)

I'm somewhat confused about the definitions of Betti numbers for Riemannian manifolds. Working with the first Betti number as an example, I have usually taken the definition to be the rank of the ...

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**2**answers

783 views

### Non-Kahler Complex manifolds

For a non-Kahler complex manifold $M$, we still have the decomposition of differential forms into differential forms of type $(p,q)$ and we can write $d=\partial+\bar\partial$ and we can define ...

**1**

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**1**answer

301 views

### When is there a deRham duality relation between the fundamental class and a top form.?

Hi, everyone:
I am reading a small expository paper on properties of CP2,
in which the intersection form is defined as an integral of
the wedge of two forms $w_1$, $w_2$, and these forms ...

**12**

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**3**answers

2k views

### A list of machineries for computing cohomology

This is not a question, but I just hope to hear more from everyone here on it.
A list of ready-to-use machineries to compute the de Rham / Cech cohomology of a manifold / variety. As far as I know, I ...

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**2**answers

1k views

### De Rham homology

Suppose M is an arbitrary smooth manifold and D is its bundle of 1-densities.
On the category of finite-dimensional vector bundles over M and linear differential operators between them
there is a ...

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votes

**8**answers

4k views

### de Rham cohomology and flat vector bundles

I was wondering whether there is some notion of "vector bundle de Rham cohomology".
To be more precise: the k-th de Rham cohomology group of a manifold $H_{dR}^{k}(M)$ is defined as the set of closed ...

**3**

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**6**answers

897 views

### Dolbeault cohomology

Hello
I am trying to get a good book that explains the Dolbeault Cohomology, does anyone know of a good one?

**9**

votes

**4**answers

748 views

### Abel's equation for the dilog

Abel's identity for the dilogarithm (see the wikipedia page about polylogarithms)
plays a role in web geometry as it is one of the abelian relations of the
first example of exceptional web (Bol's ...

**37**

votes

**2**answers

3k views

### Cohomology and fundamental classes

Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...