Let $\pi: X \rightarrow Y$ be a 2-fold branched cover of complex varieties. I know of (at least) two types of pushforwards associated to this situation:

  1. If I'm not mistaken, there is a pushforward map $\text{Nm}:\text{Pic}(X) \rightarrow \text{Pic}(Y)$ which I think is defined like this: let $\tau$ be the map on $X$ that switches sheets, and $\mathcal{L}$ a line bundle on $X$. Then define $\text{Nm}(\mathcal{L}) = (\pi_*(\mathcal{L} \otimes \tau^* \mathcal{L}))^{\tau}$ where the superscript indicates fixed points.
  2. On the other hand, we can also define a pushforward in ordinary cohomology $H^*(-, \mathbb{Z})$ as follows: Let $t: Y \rightarrow X^{\times 2}/C_2$ be defined by $t(y) = \pi^{-1}(y)$ counted with multiplicity. We then get a map on chains: $C_*(Y) \rightarrow C_*(X^{\times 2}/C_2) \rightarrow C_*(X)^{\oplus 2}/C_2 \rightarrow C_*(X)$ where the last map is addition. The dual gives a map on cochains and hence on cohomology.

My first question is, does the following diagram commute?

$\require{AMScd}$ \begin{CD} \text{Pic}(X) @>e>> H^2(X, \mathbb{Z})\\ @V \text{Nm} V V @VV \pi_! V\\ \text{Pic}(Y) @>>e> H^2(Y, \mathbb{Z}) \end{CD}

where e denotes taking the Euler class of a holomorphic line bundle.

I hope and believe the answer is yes, but I think what's getting in my way is my inability to find good references for the first construction and for branched covers in general. I imagine the proof would go by rewriting one or both of these transfers in terms of sheaf cohomology or Cech cohomology until they are visibly the same.

Even better would be a proof that constructs a norm for topological line bundles and proceeds to prove a stronger statement purely in the world of topology. So that leads me to my second question, which I should probably know the answer to: Does the norm construction for line bundles also work topologically and, more importantly, is there a reference?

  • $\begingroup$ The answer to your first question is yes. You might find it easier to find a reference for (or prove) the stronger statement that the cycle class map $\mathrm{CH}^k(X) \to H^{2k}(X)$ commutes with pushforwards for all $k$. I'm assuming the varieties are smooth. But this might not be the kind of answer you want since it basically avoids talking about line bundles, thinking instead about algebraic cycles; in particular, I don't know how to define a norm map for topological line bundles. $\endgroup$ – Dan Petersen Apr 15 '16 at 21:38
  • $\begingroup$ My hesitation here is that the pushforward I just defined for cohomology is not obviously related to the Gysin map for proper, oriented maps (which seems to be what you're claiming). Do you have a reference for that fact or am I misunderstanding? $\endgroup$ – Dylan Wilson Apr 15 '16 at 22:16
  • $\begingroup$ (For example, oriented theories usually don't have Gysin maps for ramified covers. Any cohomology theory that does splits as copies of eilenberg MacLane spectra). $\endgroup$ – Dylan Wilson Apr 15 '16 at 22:19

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