15
votes

Accepted

### Exact formula for $\chi(X, \, S^n \Omega^1_X)$

As you say, formulae for $c_1(\Omega_X^1)$ and $c_2(\Omega_X^1)$ can be obtained from the splitting principle. The following is a more general version of the calculation in this answer.
Lemma: Let $V ...

14
votes

Accepted

### Why is the first chern class of a line bundle $c_1(L) = 1-L$ in complex K-theory?

This comes from the choice of the $K$-theory Thom class for complex vector bundles.
Firstly, recall that $K$-theory $K^0(X)$ can be described as the group of bounded chain complexes of vector bundles ...

12
votes

### Chern classes of generators of $K(S^{2n})$

Note that $K(S^{2n}) = \mathbb{Z}\times\mathbb{Z}$ has many different sets of generators and these can have different Chern classes, so the question doesn't have a well-defined answer. For one ...

12
votes

Accepted

### Motivation for the definition of complex orientable cohomology theory

As you wrote, complex orientability can be characterized by the cohomology of $\mathbb{C}P^\infty$: $E$ is complex orientable if $E^*(\mathbb{C}P^\infty)$ splits according to the cell structure of $\...

11
votes

Accepted

### Chern number on non-spin manifold

The Enriques algebraic surface has even intersection form (i.e. for any class $\beta \in H^{2}(M,\mathbb{Z})$, $\int_{M^{4}} \beta^2$ is even) but is not spin by Rokhlin's theorem since the signature ...

10
votes

Accepted

### How to write K-theory Conner-Floyd Chern classes in terms of Adams operations?

I am not sure about the facts you mention, and I don't think I'll quite answer your question, but here are some facts I do know.
First, it is not the case that all $KU$-operations can be written as (...

10
votes

### Direct proof that Chern-Weil theory yields integral classes

Yes, the Chern–Weil homomorphism lifts to differential cohomology,
which guarantees that periods are integral.
See the original paper by Cheeger and Simons, or the paper by Hopkins and Singer.
The (...

10
votes

### How restrictive is having zero Chern numbers for a compact complex manifold ? Same for negative Chern number?

When a compact Kahler manifold satisfies $c_1=0$, it admits a Ricci-flat Kahler metric by Calabi-Yau, hence its tangent bundle is polystable (direct sum of stable bundles of the same slope). Then its ...

9
votes

Accepted

### Chern classes of a vector bundle

As $\mathcal{O}_{\mathbb{P}^2}$ is trivial, then multiplicativity of Chern classes in exact sequences implies:
$$
c_*(\mathcal{E}) = c_*(\mathcal{I}_p(-1)).
$$
We can compute $c_*(\mathcal{I}_p(-1))$ ...

9
votes

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

Since $H^2(S^6;\mathbb Z) = H^4(S^6;\mathbb Z) = 0$, we get $c_1(TS^6) = c_2(TS^6) = 0$. As for $c_3(TS^6)$, it equals the Euler class $e(TS^6)$, which is the Euler characteristic of $S^6$ (which ...

9
votes

Accepted

### Index of Dirac operator and Chern character of symmetric product twisting bundle

Your first question can be answered by using the splitting principle.
If $V \to X$ is a complex vector bundle of rank two, then $c_1(S^3V) = 6c_1(V)$ and $c_2(S^3V) = 11c_1(V)^2 + 10c_2(V)$.
...

8
votes

Accepted

### Is there a formula for the total Chern Class of the tangent space of a projectivized vector bundle?

No, your formula is not correct. You have to take into account the Chern classes of $V$. The relative tangent bundle $T_{\mathbb{P}V/M}$ is given by the so-called Euler exact sequence
$$0\rightarrow \...

8
votes

### A binary operation on vector bundles that adds Chern classes?

There is such an operation for $k = 2$ using virtual bundles.
Note that $c_2(E\oplus F) = c_2(E) + c_1(E)c_1(F) + c_2(F)$ so
\begin{align*}
c_2(E) + c_2(F) &= c_2(E\oplus F) - c_1(E)c_1(F)\\
&...

8
votes

Accepted

### Intersection cycle in a product of Grassmannians

Let $V$ be the $n$-dimensional space such that $\Lambda_i \subset V$. Then the condition $\dim(\Lambda_1 \cap \Lambda_2) \ge j$ is equivalent to the condition
$$
\mathrm{rank}(\Lambda_1 \...

7
votes

Accepted

### The existence of the extension of a non-trivial line bundle

This is a bordism problem, and as such can be answered using algebraic topology. I'll answer in the unoriented setting, then indicate how to modify things if $M$ and $W$ are required to be oriented.
...

7
votes

### What is the geometrical meaning of higher Chern forms and classes?

This is a big topic, which should be covered in the union of many standard texts (Chern, Griffiths-Harris, Milnor-Stasheff...). I'll list a few answers off the top of my head.
Suppose that $L$ is ...

7
votes

### What are all invariant polynomials on the space of algebraic curvature tensors?

I think this is unlikely to have a very nice answer. When $n=2$ and $n=3$, the answer is simple, but, already for $n=4$, it's not likely to be easy to give a set of generators and relations for the $\...

7
votes

Accepted

### First Chern class of torsion sheaves

The coefficient $r$ is equal to the length of $\mathcal{T}$ at the generic point of $Z$, so it is positive.

6
votes

Accepted

### 1st Chern class is invariant under choice of section?

There are many definitions of the $1$-Chern class of a complex line bundle $L\to M$, $M$ compact $CW$-complex. The topological one goes as follows. The line bundle $L$ is an oriented rank $2$ ...

6
votes

Accepted

### A binary operation on vector bundles that adds Chern classes?

Let's work with virtual bundles. Your question is equivalent to the following:
If we fix a $k \geq 1$, does the map $BU \times BU \rightarrow K(\mathbb{Z},2k)$ representing $c_k \otimes 1 + 1 \otimes ...

6
votes

Accepted

### Compactly supported chern character

Yes, this is true. For any generalized cohomology theory $E$, the compactly supported $E$-cohomology of a space $X$ is
$$E_{\mathit{cs}}^*(X) := \varinjlim\limits_{K\subseteq X:\text{ $K$ compact}} E^*...

6
votes

### Do we know any examples of complex surfaces where we have explicit knowledge of the Chern–Weil functions?

If you know the volume form, then you are asking for explicit formulas for the Chern-Weil representatives of $c_1$ and $c_2$. These would come from explicit formulas for the curvature. The calculation ...

6
votes

### How restrictive is having zero Chern numbers for a compact complex manifold ? Same for negative Chern number?

On the complex torus, all Chern numbers vanish, but the same is true on the compact complex manifold $G/\Gamma$, given by quotienting a complex Lie group by a cocompact lattice. Such lattices exist in ...

6
votes

### Coincide between Chern-connection and Levi-Civita connection

It is easier to prove this result for 1-forms, instead of vector fields. On (1,0)-forms, $\nabla^{0,1}=\bar\partial$ because the Levi-Civita connection is torsion-free, hence $\bigwedge(\nabla(\eta))=...

5
votes

Accepted

### Todd genus of symplectic $4$-manifolds a smooth invariant?

In dimension 4, the Todd genus does not depend on the choice of a symplectic structure or even on an almost complex structure. If $M$ is an almost complex 4-manifold, then $\langle c_1(M)^2, [M]\...

5
votes

Accepted

### Is there a formula for the intersection of projectivized lines inside a projectivized vector bundle?

It looks like $n_3-n_1-n_2$, but double check the computation. Tensor everything by $L_1^{-1}$ to make $L_1$ trivial and recompute the classes to get $0$, $n_2-n_1$, and $n_3-2n_1$. Then project a ...

5
votes

### Chern classes and singular hermitian metrics on vector bundles

As Hassan mentions, in the setting of singular metrics on vector bundles, the notion of curvature appears problematic, which is discussed in the paper of Raufi.
Still, as is also discussed in that ...

5
votes

### A binary operation on vector bundles that adds Chern classes?

Such an operation with values in bundles does not exist for $k = 4$ and the base space $\mathbb{HP}^2$. For virtual vector bundles, it depends how you extend the definition of the Chern classes; for ...

5
votes

Accepted

### What are all invariant polynomials on the space of algebraic curvature tensors?

I am not sure that this has a "nice" answer. Your question can be reformulated as follows. Let $\mathcal{A}_n$ be the space of algebraic curvature tensors on $\mathbb{R}^n$. A homogenous ...

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