The chern-classes tag has no wiki summary.

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### Computing the Chern class of $S^6$

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 ...

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### Chern classes of tautological bundle over the Grassmannian G(2,4) [migrated]

I've the following problem: I know how to calculate chern classes of the tautological bundle over the Grassmannian $G=G(2,4)$ using the Shubert calculus. If I am right, the chern character should be
...

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### Chern-Weil Theory for $p_1$

I'm studying Riemannian manifolds that admits a almost-complex manifold, 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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### Chern and Segre classes

I've recently started to learn about Chern and Segre classes, and it seems to me that they are very similar, sharing the same important properties and having closely related definitions.
Fulton's ...

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### Chern classes of reduced space for Hamiltonian circle action

I have a question about Chern class of symplectic reduction.
Let $(M,\omega)$ be a smooth compact symplectic manifold with a Hamiltonian circle action.
Let $H : M \rightarrow \mathbb{R}$ be the ...

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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 ...

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### top chern class

Let $E\to M$ be a holomorphic vector bundle ($M$ is compact) and assume $s\colon M\to E$ is a non-trivial section transverse to the zero section.
Is it be possible that $s^{-1}(0)\neq \emptyset$, ...

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### chern classes of push-pulled vector bundles

Let $f:X\to Y$ be a finite cover of smooth algebraic varieties, branched along a divisor $R\subset Y$. Let $E$ be a vector bundle on $Y$. What is the relation between the chern classes of $E$ and the ...

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### Preimage of $1 \in H^n(M^n)$ under Chern character

Let $M$ be a closed, oriented manifold of dimension $n$. We know that the Chern character induces an isomorphism $K^\ast(M) \otimes \mathbb{Q} \cong H^\ast(M; \mathbb{Q})$ and now I was wondering how ...

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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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### 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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### Pull-back of algebraic cycles under holomorphic maps

Let $f:X \to Y$ be a holomorphic map between two smooth complex projective manifolds. Is there a good notion of pull-back of algebraic cycles by $f$ which preserves degree in the following sense: ...

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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 $M_\alpha$, ...

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### Top chern class under finite, unramified, dominant morphism

Situation: Let $\Bbbk$ be an algebraically closed field. Assume that $\pi:Y\to X$ is an finite, dominant, unramified morphism between nonsingular varieties of dimensions $n$. Let $d=\deg(\pi)$.
What ...

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### 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 ...

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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 ...

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### About first Chern class and Poincaré duality in case of an ample divisor

Led $D$ be a very ample divisor in $X$ projective variety.
I can't understand why the first Chern class $c_1(\mathscr{O}_X(D))$ equals the PoincarĂ© dual of $D$, $\mathscr{P}(D)$

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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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### Schur polynomials in the Chern classes as direct images

Let $E\to X$ be a rank $r$ holomorphic vector bundle on a $n$-dimensional compact complex manifold. Then, it is well known that one can recover the Segre classes of $E$ as follows.
Let $\pi\colon ...

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### Short time existence on Hyperbolic Ricci flow in non-compact case

We know
Laplace equation (elliptic equations)
$ Î” u = 0$
Heat equation (parabolic equations)
$u_t âˆ’ Î”u = 0$
Wave equation (hyperbolic equations)
$u_{tt} âˆ’ Î”u = 0$
we have
- Hyperbolic geometric ...

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### sign of the First chern class fundamental group of Kahler Manifolds

We know by some facts from Kobayashi, if the Kahler manifold $M$ has positive first Chern class, i.e., $c_1 (M)>0$ then $M$ is simply connected. So if $c_1 (M)<0$ under which assumption on $M$ ...

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### 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 $c_1 (M)^2=c_1 (M)âˆ§c_1 (M)$.
So is there any recurrence ...

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### First chern class

I know some examples that first Chern class has not sign(negative, positive or zero). But I am looking for a necessary and sufficient condition that first Chern class has sign.

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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 ...

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### Does bundle with torsion Chern classes admit flat connection?

I want to know something about torsion in topological k-theory. So, consider complex bundle with chern classes lying in torsion part of integer homologies and my question is : does it admit a flat ...

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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 ...

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### (Second) Chern class of projective space, blown up in a linear subvariety

I already asked the same question at stack exchange but got no response for quite a while, so I thought I'd ask here. I also know that this has a certain resemblance to this question, but I cannot ...

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### A simple question about the degree of some vector bundle over rational curve.

Let $E$ be a holomorphic vector bundle (infact complex vector bundle is enough) over $\mathbb{P}^1$. Let $c: \mathbb{P^1} \rightarrow \mathbb{P^1}$ be the anti-holomorphic involution, ...

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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 ...

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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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### 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-Giesekerâ€™s inequality in positive characteristic", however, we can ...

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### Chern numbers of primitve classes in BU

How does one compute Chern numbers of spherical rational homology classes
$$f: S ^{2k} \to BU.$$ These generate rational homology by Milnor-Moore theorem since BU is a connected H-space, and so c_k ...

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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 ...

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### Why does the Cheeger--Chern--Simons class descend to H_3(G/B)?

Cheeger and Simons here defined a family of characteristic classes for principal $G$-bundles ($G$ a Lie group). I'm interested in the special case of $\hat c_2:H_3(SL(2,\mathbb C);\mathbb ...

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### Looking for an inequality between Chern and Todd classes (something in style of Bogomolov-Miyaoka-Yau)

Consider a smooth projective surface $S\subset\Bbb P^N_{\Bbb C}$ which is a complete intersection of hypersurfaces of degrees $(d_1,..,d_{k\ge2})$ with $d_i\ge2$ for all i. Is it true that for such ...

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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 ...

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### Inequality on Chern classes of surfaces

I remember that some where, I saw an equality like $ c_2-c_1^2 \geq 0$ on surfaces ($c_1$ and $c_2$ are Chen classes), but I don't remember the exact form of inequality neither its name.
Can you ...

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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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### 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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### 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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### 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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### Explain the relation between $K_0$ and morphisms of Chow motives

The Chern class yields an isomorphism $K_0(X)\otimes \mathbb Q\cong \bigoplus_{i\ge 0} Chow^i(X)\otimes \mathbb Q$ (for a smooth variety $X$ over a field?), whereas the latter group is isomorphic to ...

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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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### Why is the Cotangent Space of Complex Projective Space Not $U(1)$-Equivariant?

I'm looking at the cotangent bundle of $CP^{N}$ at the moment in the context of equivariance. For many reasons, it seems to me that this bundle is not $U(1)$-equivariant, or, in other words, cannot be ...

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### Tensor product of a line bundle with a large multiple of another positive line bundle also positive?

Let $X$ be a complex manifold and $\mathcal{L}$ be a positive line bundle on $X$. If $E$ is any other line bundle on $X$, then is it true that for all sufficiently large $m$, $\mathcal{L}^m \otimes E$ ...

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### Does a “Chern character” exist for any generalized cohomology theory?

The Chern character is a ring homomorphism from the complex K-theory to the usual cohomology.
1) I wonder if there are "Chern character"-like ring homomorphisms from other generalized cohomology ...

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### Proving the basic identity which implies the Chern-Weil theorem

If $E$ is a complex vector bundle over a manifold $M$ then one defines the space of vector valued $p$-differential forms on them as $\Omega^p(M,E) = \Gamma ( \wedge ^p (T^*M) \otimes E) $
The ...

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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. Let me explain where this comes from.
Suppose that $X$ is a smooth compact connected complex surface. Let $c_i=c_i(TX)$ be its $i$-th Chern ...

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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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### 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 ...