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Let $X\rightarrow \mathbb P^n_{\mathbb C}$ be a double cover ramified over a smooth hypersurface $B$ of degre $2d$. In the case of hypersurfaces of $\mathbb P^n$ one can determine the integral cohomology using Lefschetz hyperlane section theorem and universal coefficients theorem.

Q. Are there some simple techniques allowing to compute $H^k(X,\mathbb Z)$ for any $k$?

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    $\begingroup$ There is an article by Dolgachev about cohomology of weighted projective spaces. Since every cyclic, degree-$e$ cover of $\mathbb{P}^n$ branched over a hypersurface of degree $de$ is itself a hypersurface of degree $e$ in a weighted projective space $\mathbb{P}(1,\dots,1,d)$, "most" of the cohomology can be computed using this. For the primitive part, there is an extension by Carlson and Toledo of the Griffiths residue calculus to complete intersections in weighted projective space, cf. Section 4 of "Discriminant Complements and Kernels of Monodromy Representations." $\endgroup$ – Jason Starr May 30 '17 at 13:40
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If $\pi \colon X \longrightarrow Y$ is a cyclic cover of complex projective manifolds of dimension $n$ branched along an ample divisor $B \subset Y$, then by a variation of Lefschetz Hyperplane Theorem one can show that the pullback map $$\pi^* \colon H^i(Y, \, \mathbb{Z}) \longrightarrow H^i(X, \, \mathbb{Z})$$ is an isomorphism for $i \leq n-1$ and is injective for $i=n$, see Proposition 1.1 in [1].

The corresponding dual result for the push-forward map in integral homology is also true, see Proposition 1.11 in [2].

Using Poincaré duality, from this it follows that all Betti numbers of $X$ and $Y$ are equal, except possibly the middle Betti number, for which we have $b_n(X) \geq b_n(Y)$.

On the other hand, in your case we can easily find the topological Euler number $\chi_{\mathrm{top}}(X)$ by additivity. In fact the ramification locus $R \subset X$ is isomorphic to the (smooth) branch locus $S \subset \mathbb{P}^n$ and the restriction $\pi \colon X - R \longrightarrow \mathbb{P}^n - S$ is an unramified double cover, so we obtain $$\chi_{\mathrm{top}}(X) = 2(\chi_{\mathrm{top}}(\mathbb{P}^n) - \chi_{\mathrm{top}}(S))+ \chi_{\mathrm{top}}(R) = 2\chi_{\mathrm{top}}(\mathbb{P}^n) - \chi_{\mathrm{top}}(S)$$ and this allows us to recover $b_n(X)$ as well.

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References

[1] J. A. Wisniewski: On topological properties of some coverings. An addendum to a paper of Lanteri and Struppa, Can. J. Math. 44, No.1, 206-214 (1992). ZBL0766.14012.

[2] A. Lanteri, D. C. Struppa: Topological properties of cyclic coverings branched along an ample divisor, Can. J. Math. 41, No.3, 462-479 (1989). ZBL0699.14019.

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Expanding a bit Jason Starr's comment: let $R_X$ (or $R_B$) the jacobian ring of $X$ (resp. $B$). Recall that if $X=V(F) \subset w \mathbb{P}(a_0, \ldots, a_n)$, the jacobian ring is defined as $$ R_X := \mathbb{C}[x_0, \ldots, x_n]/(\partial_0(F), \ldots, \partial_n(F)),$$ with the $x_i$ suitably weighted.

From Griffiths result (and its extension to weighted projective hypersurfaces) the cohomology of $X$ can be recovered by looking at some homogeneous slices of $R_X$.

In your situation, if $B=V(f_{2d}) \subset \mathbb{P}^n$, then $X=V(y^2+f_{2d}) \subset w \mathbb{P}(1^{n+1}, d)$. In particular it is trivial to check that $R_X \cong R_B$ and you can find all the cohomology informations for $X$ by looking only at $B$. (note that in general $H^{dim \ X}(X) \neq H^{dim \ B}(B)$, since you will have to look to different homogeneous components of the Jacobian ring)

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  • $\begingroup$ Can I ask for a reference with an extension of Griffiths results to the weighted projective case? $\endgroup$ – Sasha May 31 '17 at 7:39
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    $\begingroup$ @Sasha: sure. Look at section 4 of the original Dolgachev's paper on weighted projective varieties. Theorem 4.3.2 (by Steenbrink) is the precise result. (math.lsa.umich.edu/~idolga/weighted82.pdf) $\endgroup$ – Enrico May 31 '17 at 8:28
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This is not an area of expertise for me, so forgive me if I didn't understand the question properly and hence this answer isn't on point. I think the original reference for this might be Lazarsfeld's thesis. In Springer Lecture Notes #1092, there is an article by him called "Some Applications of the Theory of Positive Vector Bundles". In section 3 of that paper he proves that for any finite map of degree d, $ f: X \to \mathbb{P}^n $ , the maps on homotopy $ f_{*} : \pi_i(X) \to \pi_i(\mathbb{P}^n) $ are bijective for $ i \leq n+1-d $ plus more. Nice paper that certainly seems relevant to your question.

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