# Questions tagged [chern-classes]

Characteristic classes associated to complex vector bundles.

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### 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 ...
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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 $X=\Tot(L)$...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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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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### 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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### 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 ...
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### 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....
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### 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 ...
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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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### 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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### 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 $E_p$...
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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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### 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 $G$...
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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 ...
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### 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 \...
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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 ...
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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 ...