Questions tagged [chern-classes]

Characteristic classes associated to complex vector bundles.

Filter by
Sorted by
Tagged with
0 votes
0 answers
60 views

Computational tasks resulting from Chern-Weil theory

I have recently learned Chern-Weil theory for smooth and complex manifolds, as well as surrounding material on cohomology with integral coefficients. I am curious what computational tasks are ...
user avatar
3 votes
0 answers
132 views

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 $X=\Tot(L)$...
Simonsays's user avatar
  • 139
4 votes
1 answer
88 views

Some questions about the definition of Chern classes in Cheeger--Simons differential characters

In page 62 to 63 of the paper "Differential characters and geometric invariants" by Cheeger and Simons, they define, among other things, Chern classes taking values in differential ...
Ho Man-Ho's user avatar
  • 1,087
2 votes
0 answers
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 ...
Mathgrad's user avatar
  • 293
1 vote
0 answers
64 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 ...
Ho Man-Ho's user avatar
  • 1,087
8 votes
0 answers
184 views

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}...
Tim's user avatar
  • 1,247
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,
maden's user avatar
  • 41
3 votes
0 answers
77 views

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, ...
Dr. Evil's user avatar
  • 2,681
3 votes
0 answers
67 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 ...
Partha's user avatar
  • 841
1 vote
0 answers
89 views

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 ...
Flavius Aetius's user avatar
1 vote
0 answers
253 views

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 ...
Tom's user avatar
  • 341
7 votes
2 answers
719 views

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^...
Tom's user avatar
  • 341
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(...
Armando j18eos's user avatar
1 vote
1 answer
189 views

Chern class of torsion sheaf support on a point

Let $X$ be a smooth projective surface. Let $p$ be a closed point of $X$. Let $k(p)$ be the corresponding skyscraper sheaf, then actually we could use Grothendieck-Riemann-Roch to calculate the Chern ...
Mike's user avatar
  • 165
1 vote
0 answers
87 views

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}) \...
Chen's user avatar
  • 1,573
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 $t\in \text{Spec}(\mathscr{O}_{F,S})$, ...
Daniel Litt's user avatar
  • 22.2k
4 votes
1 answer
663 views

Coincide between Chern-connection and Levi-Civita connection

I am a beginner in complex geometry and I am going to show Levi-Civita connection $\nabla$ and the Chern connection $D$ are the same on the holomorphic tangent bundle $T^{1,0}M$ on Kahler manifold. By ...
James Chiu's user avatar
3 votes
0 answers
248 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 ...
Hydrogen's user avatar
  • 303
7 votes
2 answers
640 views

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 ...
Irène's user avatar
  • 71
1 vote
0 answers
198 views

Comparison of first Chern class

Let $X$ be a smooth projective surface over $\mathbb{C}$, $M, N$ are rank two vector bundles (locally free sheaves of rank two) on $X$. Moreover, $N$ is a subsheaf of $M$. My first question is why $c_{...
Mike's user avatar
  • 165
1 vote
1 answer
339 views

For a vector bundle over a curve, is there a condition on the Hilbert polynomial for no non-zero section?

Assume we are over $\mathbb C$. Let $C$ be a complete algebraic curve, and $E$ an algebraic vector bundle. Its Hilbert polynomial is $$p(t)=rt+r(1-g)+d$$ where $r=\mathrm{rank}(E)$ and $d=\deg(E)$ and ...
Display Name's user avatar
8 votes
1 answer
328 views

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, \...
ChernSlope's user avatar
3 votes
0 answers
143 views

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-...
ChernSlope's user avatar
5 votes
1 answer
581 views

First Chern class of torsion sheaves

Let $X$ be a smooth projective variety, $\mathscr T$ a torsion sheaf with irreducible support of codimension $1$, say $Z$. Then the first Chern class $c_1(\mathscr T)$ is of form $r[Z]$. Is there ...
Makimura's user avatar
  • 113
5 votes
1 answer
293 views

Intersection cycle in a product of Grassmannians

Let $G(k,n)$ denote the Grassmiannian of $k$-planes in $\mathbb C^n$. Let's define $$ I_j =\{ (\Lambda_1,\Lambda_2 ) \in G(k,n) \times G(l,n) \, | \, \dim(\Lambda_1 \cap \Lambda_2) \geq j \}. $$ These ...
Blazej's user avatar
  • 334
1 vote
0 answers
264 views

Can Chern class determine nefness?

Setting: $X$ is a compact complex manifold (not necessarily Kahler, not to mention projective), suppose $L_1$ and $L_2$ are two holomorphic line bundles on $X$. Now the Chern classes $c_1(L_1)=c_1(L_2)...
Invariance's user avatar
7 votes
4 answers
837 views

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),...
Francesco Polizzi's user avatar
3 votes
1 answer
259 views

Different ways of defining the Chern character of a complex

Consider a finite complex $E$ of (holomorphic) vector bundles on a (complex) manifold $X$, i.e, the complex is of the form $$ 0 \to E_N \to E_{N-1} \to \dots \to {E_0} \to 0, $$ where the bundles are ...
Richard Lärkäng's user avatar
2 votes
1 answer
205 views

Making coherent sheaves with nonvanishing higher Chern classes

Let $\mathcal{F}$ be a coherent sheaf on a variety $X$, and assume $\mathcal{F}$ has generic rank $n$. I expect (see e.g. here) that this actually puts no conditions on its Chern classes $c_1(\mathcal{...
Pulcinella's user avatar
  • 5,506
1 vote
0 answers
154 views

A question about self-intersecting normal crossing divisors

Let $D=D_1\cup D_2$ be a simple normal crossing (snc) divisor in a smooth complex projective variety $X$. Let $E=\mathcal{O}_X(V_1)\oplus \mathcal{O}_X(V_2)$. Then, obviousely, $$ c(E)\equiv 1+c_1(E)+...
Mohammad Farajzadeh-Tehrani's user avatar
4 votes
0 answers
125 views

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 ...
GabrieleBenedetti's user avatar
7 votes
1 answer
588 views

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 ...
Tommaso Rossi's user avatar
1 vote
1 answer
228 views

Vector bundles admitting resolution by ample line bundles

Let's assume we are working a smooth projective variety. Let $C$ be the category of vector bundles constructed by taking successive extensions of line bundles of the form $\mathcal{O}(n)$ for $n\in \...
user127776's user avatar
  • 5,851
1 vote
0 answers
269 views

How to calculate the total chern classes of CP^n [closed]

When calculating the total chern class of $\ CP^n$, we use the fact that their is a exact sequence of vector bundles over $\ CP^n$: $$\ 0\to S \to C^{n+1} \to Q \to 0$$ And identify the bundle $\ TCP^...
Sunhf's user avatar
  • 157
11 votes
0 answers
253 views

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 ...
user108687's user avatar
5 votes
1 answer
294 views

Compactly supported chern character

It is a standard result that for a CW complex $X$, the chern character $$\text{ch}: K^*(X)\otimes_{\mathbb{Z}} \mathbb{Q}\to H^*(X,\mathbb{Q})$$ induces an isomorphism. Suppose now that $X$ is an open ...
Arkadij's user avatar
  • 914
3 votes
1 answer
307 views

Homotopy Ehresmann and deformation invariance of $l$-adic Chern classes

Let $S$ be a connected scheme of finite type over $\overline{\mathbb{F}_p}$. Let $\pi:X\to S$ be a smooth proper morphism such that each fiber over a closed point has a trivial étale fundamental group....
user avatar
7 votes
2 answers
394 views

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^* ...
Malkoun's user avatar
  • 5,011
7 votes
2 answers
647 views

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 =...
Xiao-Gang Wen's user avatar
6 votes
1 answer
452 views

Action of Steenrod algebra on Chern classes

This is question about result from Brown and Peterson $H^*(MO)$ as an algebra over the Steenrod algebra. Unfortunately, the paper is not available on the Internet, so I can't find the proof. One of ...
user avatar
18 votes
4 answers
1k views

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 ...
Troshkin Michael's user avatar
10 votes
1 answer
583 views

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 ...
Guest123412341234's user avatar
3 votes
1 answer
819 views

Chern classes of complex vector bundle

I'm reading characteristic classes form the book Differential forms in Algebraic Topology by Bott and Tu. The Chern classes are defined as follows: $E\xrightarrow{\rho} M$ is a vector bundle and $E_p$...
SUDEEP PODDER's user avatar
17 votes
1 answer
1k views

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 $F_A$ ...
Mohan Swaminathan's user avatar
4 votes
1 answer
267 views

Chern -Weil map for topological principal G bundles

Let $G$ be a Lie group. In the book Curvature and Characteristic classes, the author (Johan L. Dupont) mentiones in beginning of chapter 5 the following : The notion of a topological principal $G$...
Praphulla Koushik's user avatar
11 votes
3 answers
697 views

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 ...
John Greenwood's user avatar
2 votes
1 answer
179 views

Comparing the minimal Chern number and the cup-length of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. One can define its minimal Chern number $N_M$ as: $$ N_M := \text{inf} \lbrace k > 0 \ |\ \exists A \in H_2(M; \mathbb{Z}), \langle c_1, A \rangle = k \...
BrianT's user avatar
  • 1,197
4 votes
0 answers
479 views

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 ...
user127776's user avatar
  • 5,851
6 votes
0 answers
224 views

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 ...
Eric Schlarmann's user avatar
6 votes
0 answers
178 views

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 ...
Armando j18eos's user avatar