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

Question

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 characters. Let me recall part of their construction: let $V\to M$ be a complex vector bundle of rank $n$ with connection $\nabla$. Let $\pi:V_{n-k+1}\to M$ be the Stiefel bundle with fiber the Stiefel manifold $E_{n-k+1}$, which consists of complex $n-k+1$ local frames of $V\to M$. We have the following exact sequence

$$\mathbb{Z}\cong H_{2k-1}(E_{n-k+1}) \to H_{2k-1}(V_{n-k+1}) \to H_{2k-1}(M) \to 0$$ where the map $H_{2k-1}(V_{n-k+1}) \to H_{2k-1}(M)$ is denoted by $\pi_*$. Let $p:E(V)\to M$ be the principal $\textrm{GL}(n; \mathbb{C})$ frame bundle with connection $\theta$ and curvature $\Omega$. Then the pullback of the $k$-th Chern form $\pi^*(c_k(\Omega))$ (in the paper it is $\pi^*(c_i(\Omega))$, but I am not sure if it is a typo or not) is exact in $V_{n-k+1}$, and there exists a family of canonical (2k-1) forms $Q_{2k-1}$ in $V_{n-k+1}$, defined modulo exact forms, which is natural in the category of principal $G$-bundle with connections and satisfies

$$dQ_{2k-1}=\pi^*(c_k(\Omega))\textrm{ and }\int_{h_{2k-1}}Q_{2k-1}=1,$$ where $h_{2k-1}$ is a generator of $H_{2k-1}(E_{n-k+1})\cong\mathbb{Z}$.

I understand very roughly that the above should be related to the definition of the ordinary Chern classes using obstruction theory (after some googling), but I know nothing about it. My questions are:

1. Why is $\pi^*(c_k(\Omega))$ exact in $V_{n-k+1}$?

(I understand that the pullback $p^*E(V)\to E(V)$ is trivial, so $p^*(c_k(\Omega))$ is exact.) Is it because of the definition of Chern classes in terms of obstruction theory?

2. It looks to me that the canonical (2k-1) forms $Q_{2k-1}$ in $V_{n-k+1}$ is something like the transgression form $Tc_k(\theta)$ of the $k$-th Chern form $c_k(\Omega)$, which lives in the total space $E(V)$. But since the forms $Q_{2k-1}$ live in $V_{n-k+1}$, what can we say about these forms in terms of the connection $\nabla$ on $V$ or $\theta$ on $E(V)$?

Any reference related to the above is very welcome.

Answer 1

There is a general construction of a transgression form on $P/H$ when $P$ is a principal $G$ bundle with connection, $H \subset G$ is a reductive subgroup, and $f\in \text{Sym}^n(\mathfrak{g}^*)$ is an invariant polynomial which vanishes on $\mathfrak{h}$. This construction appears in https://arxiv.org/pdf/0906.3909.pdf and is due to Chern in the cases of the Euler class, and Chern classes. It was explained to me in the following way by Dan Freed.

The general Chern-Simons construction gives you a transgression form on $P$. That is, a $2n-1$ form $\alpha$ on $P$ such that

$$d\alpha = f(\Omega) $$

where $\Omega$ is the curvature of the connection, and $f$ is an invariant polynomial on the lie algebra \mathfrak{g} of degree $n$. Here, $f(\Omega)$ is referred to as the Chern-Weil form. It turns out that when $f$ vanishes on $\mathfrak{h}$, you can find a transgression form on the intermediate bundle $P/H$. As I understand it, the catch is that the standard Chern-Simons form on $P$ does not necessarily descend to $P/H$, but you can construct a transgression form by a method similar to the standard Chern-Simons method:

Note that the projection $p:P \to P/H$ is a principal $H$ bundle, which gets a connection by using the invariant projection $g \to h$ (existence of this projection is the definition of reductive). The induced $G$ bundle (via the inclusion $H \to G$) thus inherits a connection. Note that the Chern-Weil form given by applying $f$ to the curvature of this connection vanishes because $f$ vanishes on $\mathfrak{h}$. This induced G bundle turns out to naturally coincide with the pullback to $P/H$ of $P$, which has its own pullback connection. By the Chern-Simons "integration along the interval" trick we get a $2n-1$ form whose exterior derivative is the difference of the Chern-Weil forms of the two connections. Since the first Chern-Weil form vanished, its exterior derivative is simply the second, which is the pullback to $P/H$ of the Chern-Weil form of the original connection on $P$.

In the case of Chern classes, consider $U(k-1) \subset U(n)$. The lie algebra $\mathfrak{u}(k-1)$ consists of matrices with at most $k-1$ non-zero eigenvalues. This means that the Chern-Weil form for $c_k$ vanishes. We thus should be able to write down a transgression form on the associated $U(n)/U(k-1)$ bundle. Finally, note that $U(n)/U(k-1)$ is indeed the $n-k+1$ Stiefel manifold.