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18
votes
2answers
967 views

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 ...
15
votes
1answer
2k views

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 ...
13
votes
2answers
1k views

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 ...
12
votes
0answers
598 views

Poincare-Hopf and Matthai-Quillen for Chern classes?

One. The Poincare-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 ...
11
votes
3answers
3k views

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 ...
9
votes
0answers
455 views

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 ...
8
votes
1answer
394 views

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, ...
7
votes
3answers
811 views

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 ...
7
votes
1answer
466 views

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 ...
7
votes
2answers
702 views

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 ...
7
votes
3answers
1k views

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 ...
6
votes
3answers
1k views

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 ...
6
votes
1answer
225 views

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 ...
6
votes
3answers
482 views

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 $\mathfrak g\mathfrak l_n(\mathbb C)$ ...
6
votes
1answer
650 views

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 ...
6
votes
1answer
568 views

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 ...
5
votes
2answers
425 views

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 ...
5
votes
1answer
395 views

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 ...
5
votes
1answer
539 views

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 ...
5
votes
2answers
370 views

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 ...
4
votes
1answer
371 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 $H-1$ ...
4
votes
3answers
702 views

(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 ...
4
votes
1answer
254 views

Does the non-commutative Chern class depend on the choice of connection?

In classical geometry the calculation of the Chern classes of a vector bundle using a connection is independent of the choice of connection. Does any such result hold for projective modules in ...
4
votes
4answers
917 views

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 ...
4
votes
1answer
745 views

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 ...
4
votes
2answers
329 views

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 ...
4
votes
1answer
502 views

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 $m$, $\mathcal{L}^m \otimes E$ ...
4
votes
0answers
104 views

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) ...
4
votes
0answers
199 views

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
0answers
150 views

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
0answers
834 views

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
0answers
342 views

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
votes
1answer
314 views

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 ...
3
votes
1answer
315 views

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, ...
3
votes
2answers
232 views

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 ...
3
votes
1answer
124 views

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 ...
3
votes
1answer
290 views

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 ...
3
votes
1answer
322 views

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 $s^{-1}(0)\neq \emptyset$, ...
3
votes
1answer
534 views

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

We know Laplace equation (elliptic equations) $ Δ u = 0$ Heat equation (parabolic equations) $u_t − Δu = 0$ Wave equation (hyperbolic equations) $u_{tt} − Δu = 0$ we have - Hyperbolic geometric ...
3
votes
0answers
176 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 ...
2
votes
3answers
744 views

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 ...
2
votes
2answers
167 views

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 ...
2
votes
1answer
246 views

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 ...
2
votes
1answer
410 views

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 ...
2
votes
1answer
382 views

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 ...
2
votes
1answer
111 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:= ...
2
votes
1answer
137 views

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 ...
2
votes
1answer
180 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 ...
2
votes
1answer
207 views

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 ...
2
votes
0answers
68 views

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 ...