Theorem. Let $D$ be a strict normal crossings divisor on a smooth projective connected variety $X$ over $k$ of dimension at least two. 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$. Then • The singularities of$Y$occur only lie in$\pi^{-1}D^{sing}$, where$D^{sing}$is the singular locus of$D$; • We have $$e_c(Y) = \deg \pi e_c(X) + e_c(\pi^{-1}(D^{sing}))-\deg \pi e_c(D^{sing}).$$e_c(D^{sing}) + e_c(\pi^{-1}(D-D^{sing})) - \deg \pi e_c(D-D^{sing}).$$• Final Remark. In the last formula$$e_c(\pi^{-1}(D-D^{sing})) - \deg \pi e_c(D-D^{sing})= e_c(D-D^{sing})(\deg \pi - d^\prime),$$where d^\prime is the degree of the finite etale morphism \pi^{-1}(D-D^{sing})\to D-D^{sing}. (In a previous version I thought this was always zero, because I mistakingly assumed d^\prime = \deg \pi.) 5 deleted 7 characters in body Angelo's answer is complete, but I think you would be interested in the following (which is more about Euler characteristics than Chern classes). I will assume k= \mathbf C, but what I will write holds for k algebraically closed of characteristic zero once you replace "cohomology with compact support and coefficients in \mathbf Q on the category of para-compact Hausdorff topological spaces" by "etale cohomology with compact support and coefficients in \mathbf Q_\ell for some prime \ell on the category of finite type separated k-schemes". Let H^\cdot_c(-,\mathbf Q) denote cohomology with compact support and coefficients in \mathbf Q on the category of para-compact Hausdorff topological spaces. For a finite type separated \mathbf C-scheme, write e_c(X) for the Euler characteristic of X, i.e., e_c(X) = \sum_{i} (-1)^i \dim_{\mathbf Q} H^i_c(X,\mathbf Q). Since X is separated and of finite type, this is a well-defined integer. (Of course, I'm implicitly utilizing the analytification of X here.) Theorem. Let \pi:X\to Y be a finite etale morphism of finite type separated \mathbf C-schemes. Then e_c(X) = \deg \pi e_c(Y). Proof. We may and do assume X and Y are connected. Also, we may and do assume \pi:X\to Y is Galois. (In fact, let P\to Y be a Galois closure of X\to Y. Let G be the Galois group of P\to Y. Let H be the subgroup of G such that P/H = X. Then $$e_c(Y) = \frac{e_c(P)}{\# G} = \frac{\# H}{\# G} e_c(X) = \frac{1}{\deg \pi} e_c(X)$$ and so the result follows in the general case.case.) Thus, we have a finite group G acting freely (without fixed points) on Y such that X=Y/G. Note that \deg \pi = \vert G\vert. Apply the Lefschetz trace formula to see that Tr(g,H^\ast_c(Y)) =0 for any g\neq e in G. By character theory for \mathbf Q_\ell[G], we may conclude that the element$$ [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).$$The first equality follows from the Lemma, the second from the Theorem and the third from the Lemma. QED We can use this Corollary to obtain a more precise description of the "error term" under some mild hypotheses. Recall that a strict normal crossings divisor on a smooth projective variety is a divisor whose irreducible components are smooth and intersect transversally. Theorem. Let D be a strict normal crossings divisor on a smooth projective connected variety over k. k of dimension at least two. Let U be the complement of the support of D in X and let V\to U be a finite etale morphism with V connected. Let \pi:Y\to X be the normalization of X in the function field of V. Then 1. The singularities of Y are quotient singularities (and thus rational singularities); 2. The singularities of Y occur only in \pi^{-1}D^{sing}, where D^{sing} is the singular locus of D; 3. The morphism obtained by restriction \pi^{-1}(D-D^{sing})\to D-D^{sing} is etale; 4. We have$$e_c(Y) = \deg \pi e_x(Xe_c(X) + e_c(\pi^{-1}(D^{sing}))-\deg \pi e_c(D^{sing}).$$Proof. This is a long but not difficult proof. I can include the details if you'd like. For now, let me say that if you prove Y has quotient singularities, it follows that Y has rational singularities by a theorem of Viehweg; see "Rational singularities of higher dimensional schemes". To prove that Y has quotient singularities, and (1), (2) and (3) you use results from SGA1 on the fundamental group. Note that (4) follows from the Corollary, the Lemma and (3). QED 4 added 4 characters in body Angelo's answer is complete, but I think you would be interested in the following (which is more about Euler characteristics than Chern classes). I will assume k= \mathbf C, but what I will write holds for k algebraically closed of characteristic zero once you replace "cohomology with compact support and coefficients in \mathbf Q on the category of para-compact Hausdorff topological spaces" by "etale cohomology with compact support and coefficients in \mathbf Q_\ell for some prime \ell on the category of finite type separated k-schemes". Let H^\cdot_c(-,\mathbf Q) denote cohomology with compact support and coefficients in \mathbf Q on the category of para-compact Hausdorff topological spaces. For a finite type separated \mathbf C-scheme, write e_c(X) for the Euler characteristic of X, i.e., e_c(X) = \sum_{i} (-1)^i \dim_{\mathbf Q} H^i_c(X,\mathbf Q). Since X is separated and of finite type, this is a well-defined integer. (Of course, I'm implicitly utilizing the analytification of X here.) Theorem. Let \pi:X\to Y be a finite etale morphism of finite type separated \mathbf C-schemes. Then e_c(X) = \deg \pi e_c(Y). Proof. We may and do assume X and Y are connected. Also, we may and do assume \pi:X\to Y is Galois. (In fact, let P\to Y be a Galois closure of X\to Y. Let G be the Galois group of P\to Y. Let H be the subgroup of G such that P/H = X. Then $$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).$$The first equality follows from the Lemma, the second from the Theorem and the third from the Lemma. QED We can use this Corollary to obtain a more precise description of the "error term" under some mild hypotheses. Recall that a strict normal crossings divisor on a smooth projective variety is a divisor whose irreducible components are smooth and intersect transversally. Theorem. Let D be a strict normal crossings divisor on a smooth projective connected variety over k. Let U be the complement of the support of D in X and let V\to U be a finite etale morphism with V connected. Let \pi:Y\to X be the normalization of X in the function field of V. Then 1. The singularities of Y are quotient singularities (and thus rational singularities); 2. The singularities of Y occur only in \pi^{-1}D^{sing}, where D^{sing} is the singular locus of D; 3. The morphism obtained by restriction \pi^(D-D^{sing})\to \pi^{-1}(D-D^{sing})\to D-D^{sing} is etale; 4. We have$$e_c(Y) = \deg \pi e_x(X) + e_c(\pi^{-1}(D^{sing}))-\deg \pi e_c(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 that$Y\$ has quotient singularities, and (2) and (3) you use results from SGA1 on the fundamental group. Note that (4) follows from the Corollary, the Lemma and (3). QED