Questions tagged [chern-classes]
Characteristic classes associated to complex vector bundles.
62
questions with no upvoted or accepted answers
22
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958
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Poincaré-Hopf and Mathai-Quillen for Chern classes?
One. The Poincaré-Hopf theorem is usually stated as a formula for the Euler characteristic of the tangent bundle TM. Is there a version for Euler classes, of oriented real vector bundles?
It seems ...
18
votes
0
answers
2k
views
Cycles in algebraic de Rham cohomology
Let $F$ be a number field, $S$ a finite set of places, and $X$ a smooth projective $\mathscr{O}_{F,S}$-scheme with geometrically connected fibers. For each point $t\in \text{Spec}(\mathscr{O}_{F,S})$, ...
11
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0
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252
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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 ...
10
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0
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732
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What is Quillen's contribution to index theorem?
In the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne it is said that "Our book is based on a simple principle, which we learned from D. Quillen: Dirac operators are a ...
9
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0
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236
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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, ...
8
votes
0
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181
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Čech representatives for Chern classes in holomorphic Deligne cohomology
Let $X$ be a complex-analytic manifold with "nice" (e.g. Stein) cover $\mathcal{U}=\{U_\alpha\}$, and $E$ a holomorphic vector bundle on $X$ defined by transition functions $\{g_{\alpha\beta}...
8
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522
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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
$$\...
7
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288
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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$ ...
7
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0
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201
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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 ...
6
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224
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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 ...
6
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178
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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 ...
6
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325
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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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0
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92
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Elliptic deformation of the second Chern class
Second Chern class
$$c_2 \in H^4(BGL,\mathbb{Q}(2))$$
admits a nice presentation using dilogarithm. The five term relation in this setting becomes a cocycle condition (details can be found here). ...
6
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334
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Schubert cycles on Grassmannian bundles
Let $X$ be smooth variety and let $\mathcal{E}$ be a vector bundle on $X$ of rank $n$. On the total space of the Grassmannian bundle $\pi:G(k,\mathcal{E})\to X$ we have the tautological exact sequence ...
6
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360
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Why write GRR with the relative tangent sheaf?
The first published version of the Grothendieck-Riemann-Roch theorem, GRR for short, was written in the form
$$
\operatorname{ch}(f_!\alpha).\operatorname{Td}(Y)
=
f_*\left(\operatorname{ch}(\alpha).\...
5
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0
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160
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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 $\...
5
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0
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276
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Strategy to prove formula for top chern class from knowlege of chern character
I am trying to prove a conjecture that involves an enumerative problem. In the course of doing so, the following situation came up.
I have a sequence of (smooth, complex, rationally connected) ...
5
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0
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2k
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How to compute the Chern class of a projective bundle?
For example, what is the first Chern class of $X:=\mathbb{P}(T\mathbb{P}^3)$ and $Y:=\mathbb{P}(T^*\mathbb{P}^3)$?
I am asking this question because I saw an essay today by F.Hirzebruch, saying that ...
4
votes
0
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122
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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 ...
4
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479
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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 ...
4
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380
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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 ...
4
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0
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788
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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}...
4
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0
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214
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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(-\...
4
votes
0
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150
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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 ...
4
votes
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257
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What is the structure of the stack of complexes supported in dimension less than r?
Let $X$ be something. (smooth and projective variety over C are my assumptions)
The stack $M$ parameterising coherent sheaves on $X$ splits as a disjoint union of open and closed substacks $M_\alpha$, ...
4
votes
0
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197
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Inequalities between numerical invariants of nonsingular projective Varieties in positive Characteristic
It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can ...
4
votes
0
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382
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Atiyah--Singer for the Complex Projective Line
I'm trying to understand Atiyah--Singer by looking at the usual starting point of $CP^1$ and the Dirac--Dolbeault operator. If I've reduced everything down correctly, then in this case the theorem ...
3
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132
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Simple Grothendieck-Riemann-Roch computation with relative Todd class
$\DeclareMathOperator\Tot{Tot}\DeclareMathOperator\ch{ch}\DeclareMathOperator\td{td}\DeclareMathOperator\ker{ker}\DeclareMathOperator\rk{rk}$I was wondering if the following is correct: Let $X=\Tot(L)$...
3
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0
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74
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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, ...
3
votes
0
answers
66
views
Bound on norm of the curvature from Chern class
Let $M$ be an oriented closed $6$-manifold. $V$ be an hermitian complex vector bundle of dimension $4$ on $M.$ Hence $c^3(V)\in H^6(M,\mathbb{Z})\cong \mathbb{Z}$ can be thought of as an integer and ...
3
votes
0
answers
246
views
How to define Chern classes on complex analytic spaces?
Let $X$ be a complex analytic space, assume normal if needed, and $\mathscr F$ be a coherent sheaf. How to define Chern classes $c_i(\mathscr F)$? Do the usual Chern class axioms hold in this case? Is ...
3
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0
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142
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Reference request: Chern slopes
Let $M$ be a compact Kähler surface. The Chern slope is defined to be $$c_1^2/c_2,$$ where $c_1,c_2$ are the first and second Chern classes of $M$.
The classic Compact complex surfaces book by Barth-...
3
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0
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224
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Symplectic Chern class of holomorphic symplectic manifold
I've posted this question already on math.stackexchage. Unfortunately, it did not receive any answers even though there was a bounty on it. So maybe somebody of you could help me. I apologize if this ...
3
votes
0
answers
373
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The Chow ring of a blow-up along a badly embedded subscheme
Let $X$ be a variety and $i:Y\hookrightarrow X$ be a regularly embedded closed subvariety, and write $\tilde X$ for $Bl_Y X$ the blow-up of $X$ along $Y$. Let $A^*(X)$ refer to the Fulton-Macpherson ...
3
votes
0
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165
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Cycle maps as edge maps
Given a smooth projective algebraic variety over $\mathcal{C}$, let $X$ be its associated complex analytic space.
The exponential sequence on $X$:
$$0\to\mathbf{Z}(1)\to\mathcal{O}_X\to\mathcal{O}_X^...
3
votes
0
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896
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^{...
3
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0
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240
views
What is known about analogous results of Kazdan and Warner in higher dimensions?
First let me state a Theorem due to Kazdan and Warner:
``Let M be a compact two dimensional orientable manifold. Let
$f: M \rightarrow \mathbb{R}$ be a function that has the same
sign as $\chi(M)$,...
2
votes
0
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195
views
When is the Chern integral given by the norm of the curvature tensor?
I saw somewhere that for a Kahler manifold that admits a Kahler-Einstein metric the following integral formula is true.
$$\int_M c_2 \wedge \omega^{n-2} = \frac{1}{n(n-1)}\int |Rm|^2 \omega^n$$
It ...
2
votes
0
answers
92
views
Chern number of positive spinor bundle
What is the second chern number $c_2(V_+)$ of the positive spinor bundle on a 4-manifold, in particular $S^4$? Why is it that $V_+$ is the same as the quaternion line-bundle?
Thanks,
2
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304
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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
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0
answers
132
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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 $(\...
2
votes
0
answers
1k
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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), ...
2
votes
0
answers
226
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 ...
2
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0
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173
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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 ...
2
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0
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133
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degree of Chern class of logarithmic differentials
Let $X$ be a smooth complex projective variety of dimension $n$ and $D$ a normal crossings divisor. I know that the following holds:
$$
\mathrm{deg}\ c_n(\Omega^1_X(\log D))=(-1)^n \chi(U),
$$ where ...
1
vote
0
answers
62
views
The curvature of the induced connection on the antidual bundle
Let $E\to M$ be a complex vector bundle over a (real, smooth) manifold and $\nabla$ a connection on $E\to M$ whose curvature is $R$. From Section 1.5 of "Differential Geometry of Complex Vector ...
1
vote
0
answers
89
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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 ...
1
vote
0
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253
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When the whole space $H^2(X,\mathbb Q)$ can be represented by $c(L_t)$ of $X_t$?
Let $X$ be a compact complex manifold, $\pi:\mathcal X\to B$ be a holomorphic family of $X$ with $X_t=\pi^{-1}(t),t\in B$, and $X_0=X$.
Let $L_t$ be a holomorphic line bundle over $X_t$, then its ...
1
vote
0
answers
114
views
Non vanishing of a cohomology class associated to a nef vector bundle
Lemma. Let $E$ be a rank $r$ nef vector bundle over a polarized smooth complex projective variety $(X,H)$ of dimension $n\leq r$. Then for any $t\in\mathbb{R}_{\geq0}$:
$$
\sum_{k=0}^nt^{n-k}\int_Xc_k(...
1
vote
0
answers
86
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Chern class of rank one sheaves supported on subvarieties
Let $X$ be a smooth, quasi-projective variety of dimension $n$ and $\mathcal{F}$ be a globally generated coherent sheaf supported on a codimension two subvariety $V \subset X$. Is $c_2(\mathcal{F}) \...