All Questions
Tagged with chern-classes characteristic-classes
28 questions
3
votes
0
answers
79
views
Tautological ring for moduli of flat connections
Let $X$ be a smooth complex manifold and $G$ a connected complex algebraic group. Let $M$ denote the moduli stack of flat $G$-connections on $X$. Over $M\times X$, we have the tautological $G$-bundle, ...
- 2,751
2
votes
0
answers
112
views
On Brylinski–McLaughlin's paper "Čech cocycles for characteristic classes"
In the paper "Čech cocycles for characteristic classes", the authors Brylinski and McLaughlin describe how to construct Čech cocycles with values in the Deligne smooth complex representing ...
- 609
8
votes
1
answer
330
views
Do we know any examples of complex surfaces where we have explicit knowledge of the Chern–Weil functions?
Let $X$ be a compact complex surface (smooth). Let $\gamma_1, \gamma_2$ denote the Chern–Weil functions. That is, if $\omega$ is a Kähler form on $X$ with volume form $\omega^2$, then $\gamma_1, \...
- 111
3
votes
1
answer
300
views
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 ...
- 1,457
8
votes
1
answer
650
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 ...
- 641
1
vote
0
answers
292
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^...
- 157
7
votes
1
answer
477
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 ...
user131113
18
votes
4
answers
1k
views
Analogy between Stiefel-Whitney and Chern classes
There is a clear similarity between Stiefel-Whitney and Chern classes, if one replaces base field $\mathbb R$ with $\mathbb C$, coefficient ring $\mathbb Z/2$ with $\mathbb Z$ and scales the grading ...
- 481
3
votes
1
answer
927
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 $E_p$...
4
votes
1
answer
283
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 $G$...
- 6,023
12
votes
3
answers
777
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 ...
- 1,150
4
votes
0
answers
400
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 ...
- 6,023
3
votes
1
answer
544
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 ...
- 31
4
votes
0
answers
229
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
$$
ch(E,\nabla):=tr(\exp(-\...
- 9,009
3
votes
0
answers
140
views
Do we have a transgression formula for the chern characters of quasi-isomorphic cochain complexes of vector bundles?
Let $(E^{\cdot},d_E^{\cdot})$ be a cochain complex of complex vector bundles on a smooth compact manifold $X$. Now for each $E^i$ we could assign a connection $\nabla_E^i$ and obtain its curvature $(\...
- 9,009
4
votes
0
answers
152
views
tertiary characteristic class: integration of the Chern-Simons form
Let $P \to M$ be a trivial principal circle bundle with connection $A$ over a closed 3-manifold $M$. The Chern-Simons 3-form of the connection is defined by $\mathrm{CS}(A) = A \wedge dA$. Suppose ...
- 5,824
4
votes
1
answer
445
views
Chern Character Number Belongs to integer
From Getzler's definition [1], we know the odd Chern character is the following map
$$Ch:K^1(M)\to H^{odd}(M;\mathbb C),~g\mapsto \sum_{k\geqslant0}(-\frac1{2\pi\...
- 1,915
7
votes
1
answer
1k
views
Chern classes via degeneracy loci
According to book of Eisenbud-Harris Page 332 and the following summary http://pbelmans.ncag.info/blog/2014/10/09/what-are-chern-classes/
one can describe Chern classes in terms of degeneracy loci.
...
2
votes
0
answers
175
views
Chern character of finite $CW$-complexes and rational Pontrjagin class of vector bundles
Let $K$ be a finite $CW$-complex. Could you give any references or explanations for the following two items? I do not understand. Thanks!
(1). The Chern character from $\tilde{KO}^0(K)$ to the ...
- 519
5
votes
1
answer
330
views
characteristic classes of tangent bundle of 2-nd unordered configuration space
Given a (real or almost complex) manifold $M$, Let the 2-nd unordered configuration space be the quotient space
$$
B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}_2
$$
where
$$
\Delta=\{(m,m)\mid m\...
- 2,223
7
votes
0
answers
202
views
Bundles over Grassmanian with given top Chern class
So, I have been working on Chern classes for my master thesis and apparently (My proofs could be wrong and a few things are still vague) I was able to give a construction method and exhibit, via ...
- 363
5
votes
2
answers
2k
views
Is there a formula for the total Chern Class of the tangent space of a projectivized vector bundle?
Let $V\rightarrow M$ be a complex vector bundle (of rank $k$) over a complex manifold $M$ (you can assume $M$ is compact if that helps, but it may not be relevant to my question). Let $\pi:\mathbb{P}V ...
- 3,245
0
votes
1
answer
961
views
Computing the Chern class of $S^6$ [closed]
I am trying to calculate the Chern class of the tangent bundle of the sphere $S^6$. I am told that this is an interesting case, since $S^6$ is not a complex manifold, but it has an almost complex ...
- 355
9
votes
3
answers
1k
views
How does one go from Chern--Weil to cohomology classes on BGL(n,C)?
Let's assume we start with Chern--Weil theory in the following form:
Given a manifold $M$ and a complex vector bundle $V$ over $M$, we can equip $V$ with a $\mathfrak g\mathfrak l_n(\mathbb C)$ ...
- 18.7k
0
votes
1
answer
479
views
recurrence formula for *i*-th Chern class of $CP^n$
one can show that the relation between first Chern class and second Chern class of $CP^n$ is
$\frac{2(n+1)}{n} c_2 (M)=c_1 (M)^2$
here $c_1 (M)^2=c_1 (M)∧c_1 (M)$.
So is there any recurrence ...
user21574
1
vote
1
answer
3k
views
When is the first chern class of a Kaehler manifold positive/negative?
I know some examples of compact complex manifolds whose first Chern class does not have a definite sign (is neither negative, nor positive nor zero on all complex curves). I am looking for a necessary ...
user21574
25
votes
2
answers
2k
views
If the total Chern class of a vector bundle factors, does it have a sub-bundle?
Motivation: $T_{\mathbb P^2}$ isn't an extension of line bundles
Here's a trick to show that the tangent bundle $T$ of $\mathbb P^2$ is not an extension of line bundles. If it were, we would have a ...
7
votes
1
answer
2k
views
Chern classes of pushforwards
Let $f:X\to Y$ be a proper morphism of normal varieties (smooth as DM stacks, but I only care about the coarse spaces). The map $f$ is generically finite, but not flat (so no hope of smoothness and ...
- 16k