# Questions tagged [chern-classes]

Characteristic classes associated to complex vector bundles.

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**1**answer

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### 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 ...

**19**

votes

**1**answer

194 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**

votes

**1**answer

248 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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**0**answers

163 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

114 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 ...

**8**

votes

**0**answers

197 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

486 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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121 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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104 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

**1**answer

225 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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167 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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85 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 $f$ ...

**3**

votes

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47 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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66 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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votes

**0**answers

110 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**

votes

**0**answers

174 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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161 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**

votes

**0**answers

87 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 ...

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**1**answer

314 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**

votes

**1**answer

165 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

**1**answer

320 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

**1**answer

317 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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300 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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vote

**0**answers

267 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 $...

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votes

**2**answers

435 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 ...

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votes

**0**answers

393 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^{...

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130 views

### Product of Chern classes

Let $X$ be a terminal threefold
such that $K_X$ is an ample $\mathbb{Q}$-Cartier divisor
and $mK_X$ is a very ample Weil divisor.
Given $Y\in |\mathcal{O}_X(mK_X)|$ be a smooth surface,
is it possible ...

**5**

votes

**1**answer

220 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

**1**answer

414 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

143 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

**1**answer

61 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]...

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117 views

### Exact sequence for the (twisted) universal bundle of incidence correspondence

Let $p:P\rightarrow G(2,n+1)$ be the universal $\mathbb P^1$-bundle over the grassmannian. Let us denote $I=P\times_{\mathbb P^n}P$ the incidence correspondence whose generic point is of the form $([l]...

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votes

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215 views

### Injectivity of the Chern character in $K$-homology

Let $(\pi,H,F)$ be a Fredholm module: here $\pi:A \to B(H)$ is a representation of an algebra on the Hilbert space $H$ and $F$ is a self adjoint operator with square one such that for each $a \in A$ ...

**3**

votes

**1**answer

268 views

### Universal bundle of grassmannian of planes and projective bundle over grassmannian of lines

Let $p:Y=\mathbb P(\mathcal E_3^{\vee})\rightarrow G(3,n+1)$ be the universal family of hyperplanes (i.e. lines) of the planes of $\mathbb P^{n}$. The following isomorphism seems natural $$\mathcal O_{...

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votes

**1**answer

281 views

### Universal family of grassmannian as projective bundle over $\mathbb P^n$

Let $p:X=\mathbb P(\mathcal E_2)\rightarrow Gr(2,n+1)$ be the universal family of the lines in $\mathbb P^n$. If we denote $e:X\rightarrow \mathbb P^n$ the natural projection, we have $\mathcal O_{\...

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**0**answers

181 views

### Finding the Chern Class of a the pushfoward of a invertible sheaf

I am trying to understand what happens to the Chern Classes of an invertible sheaf $F$ over a complete intersection reduced curve of genus $g$ and degree $d$, when viewed as a invertible sheaf of $\...

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votes

**2**answers

626 views

### Chern classes and singular hermitian metrics on vector bundles

Let $L$ be a holomorphic line bundle on a complex manifold $X$, and assume it is equipped with a singular hermitian metric $h$ with local weight $\varphi$. Then, one can show that the de Rham class of ...

**7**

votes

**1**answer

206 views

### Chern classes of PU(n)-bundles

Let $PU(n) = U(n)/U(1)$ be the projective unitary group and denote by $BPU(n)$ its classifying space. Consider the algebra $M_n(\mathbb{C})$ as an $n^2$-dimensional Hilbert space equipped with the ...

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**0**answers

106 views

### Chern classes of a resolution of singularities

Let $j:X\subset \mathbb P_{\mathbb C}^n$ ($n\geq 3$) be a hypersurface, defined by a section of a very ample line bundle $\mathcal L$, with a ordinary double point $P$ as the only singularity and $\...

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votes

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126 views

### Flatness of Chern classes for flat family of sheaves

Let $Q$ be a quasi-projective $k$-scheme (not necessarily smooth), $X$ a smooth projective $k$-variety and $\mathcal E$ a family of (torsion free) sheaves on $X$ parametrized by $Q$. Suppose that $\...

**0**

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126 views

### Chern classes of a family and Chern classes of a member

Let $X$ be a smooth projective variety over an algebraically closed field $k$ and $\mathcal E$ a family on torsion free coherent sheaves on $X$ parametrized by a smooth curve (over $k$) i.e. a ...

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**0**answers

142 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 ...

**5**

votes

**2**answers

376 views

### 1st Chern class is invariant under choice of section?

How do I see that the 1st Chern class is invariant under choice of section? I know metric invariance follows from how two metrics on line bundle have to be conformally equivalent, but how do we show ...

**4**

votes

**1**answer

262 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\...

**6**

votes

**0**answers

186 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 ...

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**0**answers

155 views

### Chern classes, vanishing of smooth sections or vanishing of holomorphic?

I have seen both definitions and this is getting me more and more confused.
Are Chern classes dual to the degeneracy cycles of smooth sections or holomorphic?
They can't be the same thing, can they?

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votes

**0**answers

240 views

### Why write GRR with the relative tangent sheaf?

The first published version of the Grothendieck-Riemann-Roch theorem, GRR for short, was written in the form
$$
\operatorname{ch}(f_!\alpha).\operatorname{Td}(Y)
=
f_*\left(\operatorname{ch}(\alpha).\...

**7**

votes

**1**answer

2k views

### Why is the first chern class of a line bundle $c_1(L) = 1-L$ in complex K-theory?

I'm trying to understand why on earth the first chern class of a line bundle in K-theory $c_1(L) = 1-L$.
I understand that the first Chern class of the trivial bundle is zero, and that $H-1$ ...

**3**

votes

**1**answer

158 views

### Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle?

Let $E\rightarrow D$ be a complex rank two vector bundle over a compact complex
one dimensional manifold $D$. Let $L_1, L_2 \subset E$ be rank one subbundles of E
(i.e. line bundles). Let
$$ n_1:= \...

**5**

votes

**1**answer

350 views

### How does one compute the first Chern class of a Line bundle defined as the Kernel of a linear map?

Let $M$ and $N$ be compact complex manifolds of the same dimension ($m$) and
$\mu: M \rightarrow N$ a holomorphic map. Let $D \subset M$ be the subset of
points of $M$, where $d\mu|_p$ fails to be ...