Suppose $\pi:C'\to C$ is a branched cover of compact Riemann surfaces such that the associated extension of function fields is Galois with group $G$ -- so that $\pi$ presents $C$ as the quotient $C'$ by the action of $G = \text{Aut}(C'/C)$. Now, let $\rho:G\to GL(W)$ be a finite dimensional complex representation of $G$. Below, we identify locally free coherent sheaves with holomorphic vector bundles.


Let $\underline W$ be the trivial vector bundle on $C'$ with fibre $W$. Define the vector bundle $W^\rho$ on $C$ to be the subsheaf of $\pi\_\*\underline W$ whose sections over $U\subset C$ are the $G$-equivariant holomorphic functions $U' = \pi^{-1}(U)\to W$. We want to compute $c\_1(W^\rho)$ as follows. We have a natural map $\varphi:\pi^\*W^\rho\to\underline W$ on $C'$ (coming from the adjunction counit $\pi^\*\pi\_\*\underline W\to\underline W$) which is an injective map of coherent sheaves. Now, by looking at the zeros of the determinant of $\varphi$ (which occur exactly at the critical points of $\pi$), we can figure out the value of $c\_1(W^\rho) = \frac1{|G|}c\_1(\pi^\*W^\rho) = -\frac1{|G|}\cdot\dim H^0(C',\text{coker }\varphi)$.


Carrying out this computation explicitly, we seem to get the following answer. Given any branch point $p\in C$ of $\pi$, pick a preimage $p'\in C$. Let $G\_{p'}\subset G$ be the (necessarily cyclic) stabilizer group of $p'$, of order $n\_p$. For $0\le i<n\_p$, define the numbers $w\_{p,i} := \dim\text{Hom}^{G\_{p'}}((T\_{p'}C)^{\otimes i},W)$ and set $w\_p:=\sum\_{0\le i<n\_p}\frac{i}{n\_p}w\_{p,i}\in\mathbb Q$. We then get $c\_1(W^\rho) = -\sum\_p w\_p$, where the sum is over the branch points of $\pi$. **Is the result of this computation correct and can I verify it by comparing it to some well-known/basic theorem?** (I tried to calculate explicitly in local holomorphic coordinates where the map is given by $z\mapsto z^{n\_p}$ and got the above answer.)


I find it quite interesting that the sum of the **rational numbers** $w\_p$ is an **integer**. But clearly, these numbers can be defined without referring to the vector bundle $W^\rho$ or its Chern class. **Is there some direct way in which we could prove this integrality statement?**


This question is motivated by trying to understand the index computation (Theorem 4.1) in Chris Wendl's paper on super-rigidity and equivariant transversality (<https://arxiv.org/abs/1609.09867>).



Descent of vector bundle along branched cover of curve
This computation is related to the well-known semiorthogonal decomposition of the $G$-equivariant derived category of $C'$, or equivalently, of the quotient stack $C'/G$. The latter can be thought of as the curve $C$ with a root stack of order $n\_p$ structure at each of the branch point $p \in C$.


The semiorthogonal decomposition takes the form
$$
D(C'/G) = \Big\langle 
D(p\_1),\dots,D(p\_1),\dots,D(p\_m),\dots,D(p\_m),
D(C)
\Big\rangle,
$$
where the component $D(p\_i)$ repeats $n\_{p\_i}-1$ times. The exceptional objects corresponding to the components $D(p\_i)$ can be described as follows.
Let $X\_i = \pi^{-1}(p\_i)$ (taken with reduced structure). This is a subscheme of length $|G|/n\_{p\_i}$ of $C'$. Denote by $E\_{i,j}$ the sheaf $\mathcal{O}\_{X\_i}$ with $G$-equivariant structure corresponding to the action of $G$ on $T\_{x\_i}C'$, where $x\_i \in X\_i$ is any point. Then $E\_{i,1}, \dots, E\_{i,n\_{p\_i}-1}$ are the required objects.


The projection functor to the component $D(C)$ of the above semiorthogonal decomposition is equal to 
$$
F \mapsto \pi^{\ast}((\pi\_{\ast}F)^G).
$$ 
Therefore, taking $F = W$ we obtain a distingusihed triangle
$$
\pi^\ast((\pi\_\ast W)^G) \to W \to W'
$$
where $W'$ is the projection of $W$ onto the subcategory generated by $E\_{i,j}$.
Note that $(\pi\_\ast W)^G = W^\rho$. Therefore,
$$
c\_1(\pi^\ast((\pi\_\ast W)^G)) = - c\_1(W').
$$
It remains to note that $W'$ is an extension of the sheaves $E\_{i,j}$ and the integers $w\_{i,j}$ encode the multiplicities. This implies that
$$
c\_1(\pi^\ast(W^\rho) = \sum \left(jw\_{i,j} \frac{|G|}{n\_i}\right)
$$
(the last factor is the length of $E\_{i,j}$).
Finally, to obtain $c\_1(W^\rho)$ the above expression should be divided by the degree $|G|$ of the map $\pi$.


