# Questions tagged [chern-classes]

Characteristic classes associated to complex vector bundles.

137
questions

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votes

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### Different ways of defining the Chern character of a complex

Consider a finite complex $E$ of (holomorphic) vector bundles on a (complex) manifold $X$, i.e, the complex is of the form
$$
0 \to E_N \to E_{N-1} \to \dots \to {E_0} \to 0,
$$
where the bundles are ...

**2**

votes

**1**answer

165 views

### Making coherent sheaves with nonvanishing higher Chern classes

Let $\mathcal{F}$ be a coherent sheaf on a variety $X$, and assume $\mathcal{F}$ has generic rank $n$. I expect (see e.g. here) that this actually puts no conditions on its Chern classes $c_1(\mathcal{...

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86 views

### A question about self-intersecting normal crossing divisors

Let $D=D_1\cup D_2$ be a simple normal crossing (snc) divisor in a smooth complex projective variety $X$. Let $E=\mathcal{O}_X(V_1)\oplus \mathcal{O}_X(V_2)$. Then, obviousely,
$$
c(E)\equiv 1+c_1(E)+...

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59 views

### Classification of square roots of line bundles and metalinear/metaplectic structures

Reading some books and articles about geometric quantization I got confused about the classification of square roots of complex line bundles over a manifold. Consider the group of isomorphism classes ...

**6**

votes

**1**answer

381 views

### Motivation for the definition of complex orientable cohomology theory

PRELIMINARY DEFINITIONS:
Let $E^*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have:
$$
\tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt)
$$
So there is a special ...

**1**

vote

**1**answer

142 views

### Vector bundles admitting resolution by ample line bundles

Let's assume we are working a smooth projective variety. Let $C$ be the category of vector bundles constructed by taking successive extensions of line bundles of the form $\mathcal{O}(n)$ for $n\in \...

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136 views

### How to calculate the total chern classes of CP^n [closed]

When calculating the total chern class of $\ CP^n$, we use the fact that their is a exact sequence of vector bundles over $\ CP^n$:
$$\ 0\to S \to C^{n+1} \to Q \to 0$$
And identify the bundle $\ TCP^...

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183 views

### Can topological invariants be built from something different than integrals of differential forms?

I don't know whether the question is very stupid or less so, so I will give a little context, but it probably will stay too vague nonetheless.
Prelude
It is known that some topological invariants ...

**4**

votes

**1**answer

160 views

### Compactly supported chern character

It is a standard result that for a CW complex $X$, the chern character
$$\text{ch}: K^*(X)\otimes_{\mathbb{Z}} \mathbb{Q}\to H^*(X,\mathbb{Q})$$
induces an isomorphism. Suppose now that $X$ is an open ...

**3**

votes

**1**answer

277 views

### Homotopy Ehresmann and deformation invariance of $l$-adic Chern classes

Let $S$ be a connected scheme of finite type over $\overline{\mathbb{F}_p}$. Let $\pi:X\to S$ be a smooth proper morphism such that each fiber over a closed point has a trivial étale fundamental group....

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votes

**2**answers

291 views

### What are all invariant polynomials on the space of algebraic curvature tensors?

Let $V = (\mathbb{R}^n, g)$, where $g$ is the Euclidean inner product on $V$. Denote by $G$ the orthogonal group $O(V) = O(n)$ and by $\mathfrak{g}$ the Lie algebra of $G$.
Let $W \subset \Lambda^2V^* ...

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votes

**2**answers

499 views

### Chern number on non-spin manifold

Let $M^4$ be an orientable closed 4-manifold and $c_1$ be the first Chern class of a complex line bundle on $M^4$. Let $b$ be the mod 2 reduction of $c_1$, ie $b=c_1$ mod 2.
We have a relation $w_2 b =...

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**1**answer

364 views

### Action of Steenrod algebra on Chern classes

This is question about result from Brown and Peterson $H^*(MO)$ as an algebra over the Steenrod algebra. Unfortunately, the paper is not available on the Internet, so I can't find the proof.
One of ...

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**1**answer

452 views

### Index of Dirac operator and Chern character of symmetric product twisting bundle

I am having trouble understanding a couple of lines of computation from Theorem 13.30 in Besse's Einstein Manifolds text
We are twisting the spinor bundle (on Einstein 4-manifold) $\Sigma$ with an ...

**3**

votes

**1**answer

394 views

### Chern classes of complex vector bundle

I'm reading characteristic classes form the book Differential forms in Algebraic Topology by Bott and Tu. The Chern classes are defined as follows:
$E\xrightarrow{\rho} M$ is a vector bundle and $E_p$...

**15**

votes

**1**answer

687 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 $F_A$ ...

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votes

**1**answer

205 views

### Chern -Weil map for topological principal G bundles

Let $G$ be a Lie group.
In the book Curvature and Characteristic classes, the author (Johan L. Dupont) mentiones in beginning of chapter 5 the following :
The notion of a topological principal $G$...

**11**

votes

**3**answers

559 views

### A binary operation on vector bundles that adds Chern classes?

Let $E$ and $F$ be two complex vector bundles over a space $X$. There's a fairly well-known binary operation called the direct sum, written $E\oplus F$, which has the property that its first Chern ...

**2**

votes

**1**answer

118 views

### Comparing the minimal Chern number and the cup-length of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. One can define its minimal Chern number $N_M$ as:
$$
N_M := \text{inf} \lbrace k > 0 \ |\ \exists A \in H_2(M; \mathbb{Z}), \langle c_1, A \rangle = k \...

**4**

votes

**0**answers

266 views

### Chern classes of torsion-free sheaves

Let $X$ be a smooth projective variety and $Z$ a closed subvariety of co-dimension $k$. The first $k-1$ chern classes of the ideal sheaf of $Z$ vanishes and the $k$-th chern class is given by ...

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votes

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214 views

### Equivariant Venice Lemma

In the paper J. Simons and D. Sullivan. Structured vector bundles define differential K-theory, one of the key ideas is the so called Venice Lemma, which essentially can be stated as
Theorem: For ...

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113 views

### Does there exist a notion of Chern classes in intersection cohomology?

First of all: I apologize for my mistakes, I'm a freshman in intersection cohomology.
Let $X$ be a (compact) complex analytic space, let $L$ be a line bundle over $X$.
Can one define a notion of ...

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80 views

### Is chern classes of holomorphic vector bundles enough to generate Hodge cycles [duplicate]

Let $X$ ba a smooth projective variety of dimension $n$. Hodge Conjecture states that every Hodge cycle in $Hdg^k(X,\mathbb{Q})$ comes from a Chern class of codimension $k$ in $CH^k(X,\mathbb{Q})$. ...

**3**

votes

**1**answer

247 views

### Is the minimal Chern number of a toric manifold at least 2?

I would like to show that the minimal Chern number $N_M$ of a toric manifold $M$ is at least $2$, where
$$
N_M := \underset{l>0}{\min} \lbrace \exists \ A \in H_2(M;\mathbb{Z}) \ : \ \langle c, A \...

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votes

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222 views

### Can spin structures and Arf invariants be defined in terms of local quantities, like Chern classes and Chern numbers?

I'm interested in if it's possible to represent a spin structure and the Arf invariant associated to it in terms of some sort of local fields.
For example, the first Chern class of a complex line ...

**6**

votes

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69 views

### Elliptic deformation of the second Chern class

Second Chern class
$$c_2 \in H^4(BGL,\mathbb{Q}(2))$$
admits a nice presentation using dilogarithm. The five term relation in this setting becomes a cocycle condition (details can be found here). ...

**7**

votes

**1**answer

438 views

### What is the geometrical meaning of higher Chern forms and classes?

Let $M$ be a complex manifold, $R^{\nabla}$ be the curvature operator for connections $\nabla$.
Consider a polynomial function $f:\operatorname M_n(\mathbb{C})\to\mathbb{C}$. For the gauge group $\...

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votes

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308 views

### Chern-Weil theory and Weil homomorphism of principal bundle

In Kobayashi and Nomizu's book Foundations of Differential geometry they introduce the concept of connection on a principal $G$ bundle. In this book, they use connection on a principal bundle to ...

**2**

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278 views

### Topological data of $K3\times T^{2}$

I need some help in order to clarify some topological data of a $K3\times T^{2}$ Calabi Yau manifold in which $K3$ part has been obtained as a resolution of a $T^{4}/ \mathbb{Z_{2}}$ orbifold .
EDIT:...

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votes

**1**answer

148 views

### A question on the ring structure of topological K-theory and Chern character

Let $X$ and $Y$ be compact spaces (or closed manifolds if you want). I have two questions relating the ring structure of topological $K$-theory. To motivate my questions let me give some background ...

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votes

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151 views

### Symplectic Chern class of holomorphic symplectic manifold

I've posted this question already on math.stackexchage. Unfortunately, it did not receive any answers even though there was a bounty on it. So maybe somebody of you could help me. I apologize if this ...

**1**

vote

**1**answer

161 views

### determinant of curvature (notation issue)

This is when studying about Chern classes from Kobayashi and Nomizu.
Let $\pi:E\rightarrow M$ be a complex vector bundle with fibre $\mathbb{C}^r$ and Group $G=GL(r,\mathbb{C})$.
Let $p:P\rightarrow ...

**5**

votes

**1**answer

180 views

### The existence of the extension of a non-trivial line bundle

In Three Dimensional Gravity Revisted, Witten studied the Abelian Chern-Simons theory in three dimensions.
Let $W$ be a three dimensional manifold. Let $\mathcal{L}$ be a non-trivial line-bundle over ...

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votes

**1**answer

338 views

### To what extent can we characterise the image of the topological Chern character?

For a finite CW complex $X$, the Chern character gives an isomorphism
of finite-dimensional vector spaces:
$$
ch : K^*(X)\otimes \mathbb{Q} \to H^*(X, \mathbb{Q}).
$$
The vector space $V = H^*(X, \...

**9**

votes

**1**answer

438 views

### How to write K-theory Conner-Floyd Chern classes in terms of Adams operations?

From Adams, we know that the algebra of (unstable, degree-zero) cohomology operations $K^0(BU)$ can be written as formal infinite linear combinations of canonical generators
$$\mu_n := \sum_{i=0}^{n}...

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496 views

### When is the Chern class of a principal $G$-bundle same to the Chern class of the associated complex vector bundle?

Let $P\rightarrow M$ be a principal bundle, with structure group $G$. If $G$ has a representation $\rho:G\rightarrow GL(n,\mathbb{C})$, then we can define its associated vector bundle $E=P\times_{\rho}...

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vote

**1**answer

180 views

### Why is the Chern Number Invariant under A Continuously Shrinking of the Structure Group?

In Witten's paper Three Dimensional Gravity Revisited and Quantization of Chern-Simons Theory with Complex Gauge Group, he used a fact that for a principal $G$-bundle, the quantization of the Chern ...

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votes

**0**answers

344 views

### What is $SL(2,\mathbb{R})$-Chern-SImons Theory?

I found in physics that Chern-Simons theory is closely related with three dimensional gravity.
From this paper Three Dimensional Gravity Revisited, the author talks about the Chern-Simons for
$$\...

**0**

votes

**1**answer

508 views

### How do we compute the even cohomology $H^{2i}(Q)$ of the affine hyperquadric?

Consider the affine hyperquadric $Q:=\biggl\{(z_1,...,z_{n+1})\in\mathbb{C}^{n+1}\biggl|\sum_{i=1}^{n+1}z_i^2=1\biggr\}\cong TS^n$.
What is a reasonable Kähler metric for $Q$ (induced by the ...

**3**

votes

**1**answer

394 views

### Chern classes of torsion free sheaves

I am looking for the formal definition of Chern classes for torsion free sheaves, at least on the projective spaces. There are several books that do made the defintion for the vector bundle case, and ...

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votes

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233 views

### Schubert cycles on Grassmannian bundles

Let $X$ be smooth variety and let $\mathcal{E}$ be a vector bundle on $X$ of rank $n$. On the total space of the Grassmannian bundle $\pi:G(k,\mathcal{E})\to X$ we have the tautological exact sequence ...

**3**

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**1**answer

338 views

### Chern classes of generators of $K(S^{2n})$

Calculate the Chern classes $$ c_n \in H^{2n}(S^{2n})$$ for the generator of the group $$ K(S^{2n})$$ where $S^{2n} $ - sphere of dimension $ 2n $, $ K(S^{2n})$ - group from K-theory.
I found the ...

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**0**answers

240 views

### The Chow ring of a blow-up along a badly embedded subscheme

Let $X$ be a variety and $i:Y\hookrightarrow X$ be a regularly embedded closed subvariety, and write $\tilde X$ for $Bl_Y X$ the blow-up of $X$ along $Y$. Let $A^*(X)$ refer to the Fulton-Macpherson ...

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87 views

### $ch(L f^*\epsilon)$

I know the famous principle (or theorem depending on how you define) of Chern classes for locally free sheaves and a morphism $f: X'\rightarrow X$,
$ch(f^* \epsilon)=f^* ch(\epsilon)$.
But if $f$ ...

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votes

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136 views

### Will the transgression formula for superconnections give back the transgression formula of connections?

Let $E$ be a vector bundle on a smooth manifold $X$ and $\nabla$ be a connection on $E$, by Chern-Weil theory, the Chern character of $(E,\nabla)$ could be construct as
$$
ch(E,\nabla):=tr(\exp(-\...

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votes

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108 views

### Do we have a transgression formula for the chern characters of quasi-isomorphic cochain complexes of vector bundles?

Let $(E^{\cdot},d_E^{\cdot})$ be a cochain complex of complex vector bundles on a smooth compact manifold $X$. Now for each $E^i$ we could assign a connection $\nabla_E^i$ and obtain its curvature $(\...

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126 views

### Cycle maps as edge maps

Given a smooth projective algebraic variety over $\mathcal{C}$, let $X$ be its associated complex analytic space.
The exponential sequence on $X$:
$$0\to\mathbf{Z}(1)\to\mathcal{O}_X\to\mathcal{O}_X^...

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247 views

### First Chern class of the universal bundle

Let $M$ be the moduli space of semistable vector bundles of fixed determinant $L$ and rank $r$ over a smooth curve $X$. Assume that $gcd(r,deg(L))=1$. Let $\mathcal U$ be the universal bundle over $M\...

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198 views

### Chern Classes or Chern character classes in the Lichtenbaum-Quillen conjecture?

Let $F$ be a number field, $\mathcal{O}$ its ring of integers, $r>1$ an integer and $\ell$ a prime number different from $2$.
The Lichtenbaum-Quillen conjecture, now a theorem by Voevodsky, Rost, ...

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115 views

### tertiary characteristic class: integration of the Chern-Simons form

Let $P \to M$ be a trivial principal circle bundle with connection $A$ over a closed 3-manifold $M$. The Chern-Simons 3-form of the connection is defined by $\mathrm{CS}(A) = A \wedge dA$. Suppose ...