Questions tagged [chern-classes]

Characteristic classes associated to complex vector bundles.

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4
votes
1answer
213 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 ...
9
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1answer
385 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
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1answer
262 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 $E_p$...
14
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1answer
507 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 $F_A$ ...
4
votes
1answer
191 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
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3answers
528 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
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1answer
89 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
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0answers
175 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 ...
6
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0answers
208 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 ...
5
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0answers
93 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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0answers
79 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
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1answer
209 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 \...
6
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0answers
193 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
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0answers
62 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
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1answer
371 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 $\...
4
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0answers
284 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 ...
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0answers
266 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:...
2
votes
1answer
132 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 ...
3
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129 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
1answer
154 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 ...
4
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1answer
153 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 ...
20
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1answer
317 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, \...
7
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1answer
385 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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370 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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1answer
163 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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304 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 $$\...
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1answer
502 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
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1answer
319 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 ...
6
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0answers
209 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
votes
1answer
310 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 ...
3
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0answers
217 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 ...
0
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0answers
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 $f$ ...
4
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0answers
110 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(-\...
2
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0answers
94 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 $(\...
3
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0answers
125 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^...
4
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0answers
217 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\...
9
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0answers
181 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, ...
4
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0answers
111 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 ...
1
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1answer
421 views

Self-intersection of divisors and Chern class

Let $X$ be a smooth, projective variety and $Y \subset X$ a smooth, effective divisor. Consider now the natural map $i^\ast:H^2(X) \to H^2(Y)$. Then, When is the image of $c_1(\mathcal{O}_X(Y)) \in H^...
8
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1answer
177 views

Todd genus of symplectic $4$-manifolds a smooth invariant?

Suppose that $(M_{1},\omega_{1})$ and $(M_{2},\omega_{2})$ are compact symplectic $4$-manifolds, that are (oriented) diffeomorphic. Is it true that the Todd genus ($\frac{1}{12} (c_{1}^{2} + c_{2})(M_{...
5
votes
1answer
416 views

Chern classes of a vector bundle

Let $\mathcal{E}$ be a rank two vector bundle on $\mathbb{P}^2$ fitting in the following exact sequence $$0\rightarrow \mathcal{O}_{\mathbb{P}^2}\rightarrow \mathcal{E}\rightarrow \mathcal{I}_p(-1)\...
4
votes
1answer
364 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\...
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0answers
536 views

Chern Classes: two approaches

The following question is closely related to this one. Let $X$ a non singular projective variety over $\mathbb C$, and let $\mathscr E$ a locally free sheaf of rank $r$ (an algebraic vector bundle), ...
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0answers
351 views

First chern class of vector bundle w/ push / pull

(Context: I am trying to prove the Pluecker formula.) Let $E^{k+1}_L = \pi_{2,*}(\pi_1^*L\otimes O_{C\times C}/J_\Delta^{k+1})$, where $C$ is a smooth curve of genus $g$, $L$ a line bundle of degree $...
6
votes
2answers
498 views

Reference request: an example of Bott residue formula's usage

Could you give me an example of a clear and beautiful application of Bott residue formula in torus-equivariant cohomology (see below)? I found an example calculating a product of Chern classes on ...
3
votes
0answers
565 views

How does one compute the Chern classes of the dual sheaf from the Chern class of the original sheaf?

Let $X$ be a smooth projective 4-fold (over $\mathbb{C}$). Let $Z \subset X$ be a codimension two subscheme. Let $I_{Z}$ denote the ideal sheaf of $Z$. How does one compute the Chern classes of $I_Z^{...
5
votes
1answer
240 views

Reference for $E_{\infty}$-ness of the Chern Character

I would like a reference/proof for the fact that the Chern character map: $$KU_{\mathbb{Q}} \rightarrow H\mathbb{Q}[u, u^{-1}]$$ is an $E_{\infty}$-ring map. Thank you in advance!
6
votes
1answer
623 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
0answers
158 views

Computing higher dimensional intersection numbers for complete intersections of $\mathbb P^n$

Let $X_1,X_2$ be two smooth hypersurfaces of degree $d$ in $\mathbb P^{n}$. Let $B=X_1\cap X_2$. Assume $B$ is smooth. Let $\mathcal N_{B/\mathbb P^n}$ be the normal bundle to $B$. Let $H$ be the ...
0
votes
1answer
63 views

The degree of the locus of lines admitting an osculating plane

Let $Q\subset \mathbb P_k^3$ be a smooth quadric. Its Fano variety of lines $F(Q)$ is the union of two disjoints lines of $G(2,4)$ and according to the answer to this question the locus $F_{osc} =\{[l]...