16
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
Analogy between Stiefel-Whitney and Chern classes
Here is one way I like to think of the analogy.
The maximal torus of diagonal matrices $T^{n} \subset U(n)$ gives a map $BT^n \to BU(n)$ which on integral cohomology gives an isomorphism from $H^...
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
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
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 ...
8
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.
...
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
$$
\mathrm{rank}(\Lambda_1 \...
7
votes
A binary operation on vector bundles that adds Chern classes?
There is such an operation for $k = 2$ using virtual bundles.
Note that $c_2(E\oplus F) = c_2(E) + c_1(E)c_1(F) + c_2(F)$ so
\begin{align*}
c_2(E) + c_2(F) &= c_2(E\oplus F) - c_1(E)c_1(F)\\
&...
7
votes
Analogy between Stiefel-Whitney and Chern classes
Any rank $n$ real vector bundle $E\to X$, $X$ compact $CW$-complex is $\newcommand{\bZ}{\mathbb{Z}}$ is $\bZ/2$-oriented and, as such it has a $\bZ/2$-Thom class $\tau_E\in H^n_{cpt}(E,\bZ/2)$. Then
$...
7
votes
What are all invariant polynomials on the space of algebraic curvature tensors?
I think this is unlikely to have a very nice answer. When $n=2$ and $n=3$, the answer is simple, but, already for $n=4$, it's not likely to be easy to give a set of generators and relations for the $\...
7
votes
Accepted
First Chern class of torsion sheaves
The coefficient $r$ is equal to the length of $\mathcal{T}$ at the generic point of $Z$, so it is positive.
7
votes
Coincide between Chern-connection and Levi-Civita connection
It is easier to prove this result for 1-forms, instead of vector fields. On (1,0)-forms, $\nabla^{0,1}=\bar\partial$ because the Levi-Civita connection is torsion-free, hence $\bigwedge(\nabla(\eta))=...
6
votes
Accepted
A binary operation on vector bundles that adds Chern classes?
Let's work with virtual bundles. Your question is equivalent to the following:
If we fix a $k \geq 1$, does the map $BU \times BU \rightarrow K(\mathbb{Z},2k)$ representing $c_k \otimes 1 + 1 \otimes ...
6
votes
Accepted
Compactly supported chern character
Yes, this is true. For any generalized cohomology theory $E$, the compactly supported $E$-cohomology of a space $X$ is
$$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 ...
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
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
Analogy between Stiefel-Whitney and Chern classes
Let me try a high-brow answer using equivariant stable homotopy theory.
By the stable Thom isomorphism, the integral (co)homology of $BU$ agrees with that of $MU$; likewise the $\mathbb{Z}/2$-(co)...
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 ...
5
votes
Exact formula for $\chi(X, \, S^n \Omega^1_X)$
Just for reference: using the Schubert2 package from Macaulay2 this can be quite effortlessly done.
...
5
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 ...
5
votes
Accepted
When Atiyah class and Chern class coincide?
$\def\ZZ{\mathbb{Z}}\def\CC{\mathbb{C}}\def\cO{\mathcal{O}}$To spell out my comment a little more, let $Z^1$ be the sheaf of $\partial$-closed holomorphic $(1,0)$-forms. Since "holomorphic" ...
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