# Tag Info

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### 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 ...
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### 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 (...

### 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 (...

### 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 ...
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### 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 $c_*(\mathcal{I}_p(-1))$ ...

### 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 ...
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### 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 $c_2(S^3V) = 11c_1(V)^2 + 10c_2(V)$. ...
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### 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. ...

### 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 ...

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### 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^*...

### 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 ...

### 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 ...

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### 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 ...

### 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 ...
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 ...
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 ...