Questions tagged [chern-classes]
Characteristic classes associated to complex vector bundles.
166
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Does a "Chern character" exist for any generalized cohomology theory?
The Chern character is a ring homomorphism from complex K-theory to the usual cohomology.
1) I wonder if there are "Chern character"-like ring homomorphisms from other generalized cohomology theories ...
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2
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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 ...
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GAGA and Chern classes
My question is as follows.
Do the Chern classes as defined by Grothendieck for smooth projective varieties coincide with the Chern classes as defined with the aid of invariant polynomials and ...
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1
answer
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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, \...
- 1,404
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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 ...
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3
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Why is the integral of the second chern class an integer?
I'm currently reading the paper "Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase" by Barry Simon.
Imagine a vector bundle with a connection $\nabla$. For simplicity, we assume that this is ...
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4
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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 ...
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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 $t\in \text{Spec}(\mathscr{O}_{F,S})$, ...
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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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Complex vector bundles with trivial Chern classes on k-tori
Let $E\to X$ be a principal $U(N)$-bundle over a (nice) topological space $X$. It is well known that vanishing of the Chern classes of $E$ is not a sufficient condition for $E$ to be trivial, the ...
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Which torsion classes in integral cohomology are Chern classes of flat bundles?
Chern-Weil theory tells us that the integral Chern classes of a flat bundle over a compact manifold (i.e. a bundle admitting a flat connection) are all torsion. Given a compact manifold $M$ whose ...
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3
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Chern classes of a blow-up at a point
Let $X$ be a nonsingular projective variety over $\mathbb{C}$, and let $\widetilde{X}$ be the blow-up of X at a point $p\in X$.
What relationships exist between the degrees of the Chern classes of $X$...
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1
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Chern-Weil Theory for $p_1$
I'm studying Riemannian manifolds that admit an almost-complex structure, thus
$$3\tau+2\chi=c_1^2,$$
where $\tau$ is the signature, $\chi$ is the euler characteristic and $c_1$ is the first Chern ...
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3
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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 ...
- 1,131
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How does $f_* O_X$ measure ramification and Grothendieck-Riemann-Roch
Let $f:X\longrightarrow Y$ be a finite morphism of smooth projective varieties over a field $k$ of characteristic zero, where $\dim X=\dim Y$. Then $f$ is flat. Hence $f_\ast \mathcal{O}_X$ is a ...
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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 ...
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Where does the splitting principle come from and does it generalize
Basically, I'm aware of "splitting principles" for the following three objects (which are all isomorphic modulo torsion).
1. The Chow group a la Fulton.
2. The classical Grothendieck group of ...
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Can one bound the Todd class of a 3-dimensional variety polynomially in c_3
This question is on bounding the degree of the Todd class on a complex threefold.
Let $X$ be a smooth compact connected complex surface. Let $c_i=c_i(TX)$ be its $i$-th Chern class. Recall the ...
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on chern classes and Riemann Roch theorem for torsion-free sheaves on singular (possibly multiple) curve
I'm looking for a definition of Chern class (at least the first one) for a torsion-free sheaf $F$ (not necessarily locally free) on a singular curve (for simplicity can assume all the singularities ...
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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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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 ...
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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 ...
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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$ ...
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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)$ ...
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Is there a natural form representing the Thom class of a vector bundle, which when pulled back via the zero section represents the Euler class on the level of forms?
Let $V \rightarrow M$ be an oriented vector bundle over a compact
oriented manifold $M$ equipped with a metric $h$ (the metric $h$
is a metric on the Vector bundle $V$, not on the manifold $M$).
Is ...
- 3,235
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Top chern class in positive characteristic
Given a nonsingular, projective variety $X$ of dimension $n$ over an algebraically closed field $k$.
Over $k=\mathbb{C}$, the top chern class $c_n(T_X)$ of the tangent sheaf is the Euler ...
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Is there a simplicial volume definition of Chern Simons invariants?
Suppose we have some compact hyperbolic 3-manifold $M=\Gamma\backslash\mathbb H^3$. Now we know that the hyperbolic volume of $M$ can be defined as (a constant times) the simplicial volume of the ...
- 18.3k
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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, ...
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Does there always exist a line bundle whose Chern class represents an integer symplectic form?
Let $(M, \omega, J)$ be a compact symplectic manifold with a
compatible almost complex structure $J$, such that the symplectic
form determines an integer cohomology class, ie
$$ [\omega] \in H^2(M, ...
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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, \...
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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_{...
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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}...
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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
$$\...
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4
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Exact formula for $\chi(X, \, S^n \Omega^1_X)$
I have a smooth, compact complex surface $X$, and I need an explicit formula for the Euler characteristic
$$\chi(X, \, S^n \Omega^1_X),$$
where $S^n$ denotes the symmetric product, in terms of $c_1(X),...
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How restrictive is having zero Chern numbers for a compact complex manifold ? Same for negative Chern number?
In complex dimension $2$, if a surface $S$ is a blowup of a surface $S'$, one has the following relation between their Chern numbers :
$c_1^2(S) + 1 = c_1^2(S')$
$c_2(S) - 1 = c_2(S')$
By using this ...
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Chern number on non-spin manifold
Let $M^4$ be an orientable closed 4-manifold and $c_1$ be the first Chern class of a complex line bundle on $M^4$. Let $b$ be the mod 2 reduction of $c_1$, ie $b=c_1$ mod 2.
We have a relation $w_2 b =...
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Motivation for the definition of complex orientable cohomology theory
PRELIMINARY DEFINITIONS:
Let $E^*$ be a multiplicative generalized cohomology theory. By the suspension isomorphism we have:
$$
\tilde{E^2}(S^2)\cong\tilde{E^0}(S^0)=E^0(pt)
$$
So there is a special ...
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2
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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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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 ...
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2
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725
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When Atiyah class and Chern class coincide?
Let $X$ be a compact complex manifold, $L$ be a holomorphic line bundle on $X$, then the exponential exact sequence $0\to \mathbb Z\hookrightarrow \mathcal O\to \mathcal O^*\to 0$ induces the map $c:H^...
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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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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.
...
7
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1
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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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Calculating chern numbers yields a contradiction, why?
I am really stuck on this one. Let $Y=\mathbb{P}^n$ be the complex projective space and let $\tilde Y$ be the blow-up of $Y$ along a linear subvariety $X$ of codimension $d$. We get the following blow-...
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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 $a \in A$ ...
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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 ...
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votes
2
answers
428
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Chern classes of the sheaf of LOG differentials
Let $\Omega_X^1(\log D)$ be the (locally free) of logarithmic differentials on a smooth projective variety $X$ with respect to a simple normal crossing divisor $D$.
What are the Chern classes of $\...
6
votes
2
answers
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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 ...
- 1,990
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1
answer
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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 ...
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6
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2
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Uses of the Chern--Connes Pairing?
The backbone of Connes' approach to noncommutative geometry is the Chern--Connes pairing. By discovering the cyclic homology of an algebra and then pairing it the $K$-theory of that algebra, Connes ...
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