Top chern class under finite, unramified, dominant morphism

Question

Situation: Let $\Bbbk$ be an algebraically closed field. Assume that $\pi:Y\to X$ is an finite, dominant, unramified morphism between nonsingular varieties of dimensions $n$. Let $d=\deg(\pi)$.

What I know: For $\Bbbk=\mathbb C$, the second chern class of $X$ equals its topological Euler characteristic (i.e., the Euler characteristic with respect to the topology of a complex manifold). I know this under the name Gauss-Bonnet Formula. It then follows that $c_n(Y)=d\cdot c_n(X)$ because $\pi$ is a $d$-fold covering map of complex manifolds.

My Question: Does $c_n(Y)=d\cdot c_n(X)$ hold under the more general assumption that $\Bbbk$ is algebraically closed? In particular, does this hold in positive characteristic?

PS: I am mostly interested in the case $n=2$, i.e. $\pi$ is a covering map of surfaces. However, I felt that this would probably work for any $n$.

Answer 1

Yes, it does hold in positive characteristic. You can show that the degree of $c_n(X)$ equals its Euler characteristic with respect to étale cohomology with coefficients in $\mathbb{Q}_{\ell}$, where $\ell$ is a prime different from the characteristic ( Top chern class in positive characteristic ), and this is has the required behavior under étale covers ( Behaviour of euler characteristics in characteristic p for finite etale covers ).

[Edit:] the equality can be proved directly; in fact the proof is much easier. Since $\pi\colon Y\to X$ is étale, the tangent bundle $T_Y$ is the pullback $\pi^*T_X$. By functoriality of Chern classes, $c_n(Y) = \pi^*c_n(X)$ (at the level of Chow groups). But it easy to see that the composite $\pi_*\pi^*$ from the Chow group of $X$ to itself is just multiplication by $d$.

Answer 2

Angelo's answer is complete, but I think you would be interested in the following (which is more about Euler characteristics than Chern classes).

I will assume $k= \mathbf C$, but what I will write holds for $k$ algebraically closed of characteristic zero once you replace "cohomology with compact support and coefficients in $\mathbf Q$ on the category of para-compact Hausdorff topological spaces" by "etale cohomology with compact support and coefficients in $\mathbf Q_\ell$ for some prime $\ell$ on the category of finite type separated $k$-schemes".

Let $H^\cdot_c(-,\mathbf Q)$ denote cohomology with compact support and coefficients in $\mathbf Q$ on the category of para-compact Hausdorff topological spaces. For a finite type separated $\mathbf C$-scheme, write $e_c(X)$ for the Euler characteristic of $X$, i.e., $e_c(X) = \sum_{i} (-1)^i \dim_{\mathbf Q} H^i_c(X,\mathbf Q)$. Since $X$ is separated and of finite type, this is a well-defined integer. (Of course, I'm implicitly utilizing the analytification of $X$ here.)

Theorem. Let $\pi:X\to Y$ be a finite etale morphism of finite type separated $\mathbf C$-schemes. Then $e_c(X) = \deg \pi e_c(Y)$.

Proof. We may and do assume $X$ and $Y$ are connected. Also, we may and do assume $\pi:X\to Y$ is Galois. (In fact, let $P\to Y$ be a Galois closure of $X\to Y$. Let $G$ be the Galois group of $P\to Y$. Let $H$ be the subgroup of $G$ such that $P/H = X$. Then $$e_c(Y) = \frac{e_c(P)}{\# G} = \frac{\# H}{\# G} e_c(X) = \frac{1}{\deg \pi} e_c(X)$$ and so the result follows in the general case.)

Thus, we have a finite group $G$ acting freely (without fixed points) on $Y$ such that $X=Y/G$. Note that $\deg \pi = \vert G\vert$. Apply the Lefschetz trace formula to see that $Tr(g,H^\ast_c(Y)) =0$ for any $g\neq e$ in $G$. By character theory for $\mathbf Q_\ell[G]$, we may conclude that the element $$ [H^\ast_c(Y,\mathbf Q_\ell)] := \sum (-1)^i [ H^i_c(Y,\mathbf Q_\ell)]$$ in the Grothendieck group $K_0(\mathbf Q_\ell[G])$ of finitely generated $\mathbf Q_\ell[G]$-modules is given by an integer multiple of $[\mathbf Q_\ell[G]]$; the class of the regular representation. So we may write $$[H^\ast_c(Y,\mathbf Q_\ell)] = m [\mathbf Q_\ell[G]],$$ where $m\in \mathbf Z$. Now, note that $H^i_c(X,\mathbf Q_\ell) = \left(H^i_c(Y,\mathbf Q_\ell)\right)^G$ for any $i\in \mathbf Z$. Therefore, we have that $$ [H^\ast_c(X,\mathbf Q_\ell)] = m$$ in $K_0(\mathbf Q_\ell[G])$. In particular, we see that $e_c(X) = \dim_{\mathbf Q_\ell} [H^\ast_c(X,\mathbf Q_\ell)] = m$. We conclude that $$e_c(Y) = \dim_{\mathbf Q_\ell} [H^\ast_c(Y,\mathbf Q_{\ell})]= m \vert G\vert = e_c(X) \vert G \vert = \deg \pi e_c(X). $$ QED.

For completeness, here is what you can do for "ramified covers". Not surprisingly, the same equality holds up to a "correction term" coming from the branch locus.

Lemma. Let $M$ be a finite type separated $\mathbf C$-scheme. Let $N$ be a closed subscheme of $M$. Then $e_c(M) = e_c(N) + e_c(M\backslash N)$.

Proof. Mayer-Vietoris. QED

Corollary. Let $\pi:X\to Y$ be a finite flat surjective morphism, and let $D$ be a closed subscheme of $Y$ such that $\pi$ is etale over $Y\backslash D$. Then $$e_c(X) = \deg \pi e_c(Y) + e_c(\pi^{-1}D) - \deg\pi e_c(D) .$$

Proof. Write $U=Y\backslash D$ and $V=\pi^{-1}(U)$. Then $$e_c(X) = e_c(V) + e_c(\pi^{-1}D) = \deg \pi e_c(U) + e_c(\pi^{-1}D) = \deg \pi(e_c(Y) - e_c(D)) + e_c(\pi^{-1}D).$$ The first equality follows from the Lemma, the second from the Theorem and the third from the Lemma. QED

We can use this Corollary to obtain a more precise description of the "error term" under some mild hypotheses. Recall that a strict normal crossings divisor on a smooth projective variety is a divisor whose irreducible components are smooth and intersect transversally.

Theorem. Let $D$ be a strict normal crossings divisor on a smooth projective connected variety $X$ over $k$. Let $U$ be the complement of the support of $D$ in $X$ and let $V\to U$ be a finite etale morphism with $V$ connected. Let $\pi:Y\to X$ be the normalization of $X$ in the function field of $V$. Then

1. The singularities of $Y$ are quotient singularities (and thus rational singularities);
2. The singularities of $Y$ lie in $\pi^{-1}D^{sing}$, where $D^{sing}$ is the singular locus of $D$;
3. The morphism obtained by restriction $\pi^{-1}(D-D^{sing})\to D-D^{sing}$ is etale;
4. We have $$e_c(Y) = \deg \pi e_c(X) + e_c(\pi^{-1}(D^{sing}))-\deg \pi e_c(D^{sing}) + $$ $$e_c(\pi^{-1}(D-D^{sing})) - \deg \pi e_c(D-D^{sing}).$$

Proof. This is a long but not difficult proof. I can include the details if you'd like. For now, let me say that if you prove $Y$ has quotient singularities, it follows that $Y$ has rational singularities by a theorem of Viehweg; see "Rational singularities of higher dimensional schemes". To prove (1), (2) and (3) you use results from SGA1 on the fundamental group. Note that (4) follows from the Corollary, the Lemma and (3). QED

Final Remark. In the last formula $$e_c(\pi^{-1}(D-D^{sing})) - \deg \pi e_c(D-D^{sing})= e_c(D-D^{sing})(\deg \pi - d^\prime),$$ where $d^\prime$ is the degree of the finite etale morphism $\pi^{-1}(D-D^{sing})\to D-D^{sing}$. (In a previous version I thought this was always zero, because I mistakingly assumed $d^\prime = \deg \pi$.)