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If $X,Y$ are smooth projective schemes, then if we have a surjection $f:X\to Y$, we have an injective map on étale cohomology, or more generally on any Weil cohomology (see https://mathoverflow.net/q/172527). The proof of this statement uses Poincare duality on the target $Y$.

I would like to understand the cohomology of the resolution of singularity of a scheme $X$. If $\tilde{X}\to X$ is a resolution of singularites, then this is a surjective map (since it is a birational map between projective schemes). Can this be generalized to say that $$H^*(X)\to H^*(\tilde{X})$$ is injective? Since the ``usual" argument uses Poincare duality on the target $X$, and $X$ is non-smooth (otherwise, what's the point of the resolution of singularities), the usual argument fails.

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    $\begingroup$ No, but it's true for intersection cohomology. $\endgroup$ – David Hansen Dec 12 '20 at 20:46
  • $\begingroup$ @DavidHansen Great, thank you so much! $\endgroup$ – curious math guy Dec 12 '20 at 22:04
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In general, the answer is no. It already fails for a nodal curve. In fancier terms, you can understand the obstruction as follows: if $X$ has a resolution $\tilde X$, and the cohomology of $X$ injects the cohomology of $\tilde X$, then $H^i(X)$ would be pure of weight $i$ as a Galois module/mixed Hodge structure (when over $\mathbb{C}$).

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Given a resolution $\pi:\tilde X \to X$, you can ask whether the pullback morphism $\pi^*:H^k(X) \to H^k(\tilde X)$ is injective for some (or all) $k$. As Donu points out, the mixed Hodge structure on $H^k(X, \mathbf C)$ would be pure which is a restrictive condition. One case where this is true for all $k$ is when $X$ has at worst quotient singularities by a result of Steenbrink. For orbifolds, $H^k(X) \cong IH^k(X)$ for all $k$, where $IH^k(X)$ is the intersection cohomology, so we have the injection again by David's comment. Another example for a specific $k$ is for rational singularities, i.e., $R^i\pi_*\mathcal O_{\tilde X} = 0$ for all $i > 0$. By playing around with the Leray spectral sequence, it follows that $\pi^*$ is injective for $k \le 2$ (although I think $H^1(X)$ is pure for normal singularities by the same logic). In general $H^2(X)$ and $IH^2(X)$ are not isomorphic for rational singularities. You can see this by taking a scheme with a isolated rational singularity which is not $\mathbf Q$-factorial.

However, there is always a map $H^2(X) \to IH^2(X)$ which will be injective if $H^2(X)$ carries a pure Hodge structure. More generally, any injection $H^k(X) \hookrightarrow H^k(\tilde X)$ will factor through the injection $IH^k(X) \hookrightarrow H^k(\tilde X)$ coming from the decomposition theorem (at least for proper schemes).

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  • $\begingroup$ Do you have a reference for this result of Steenbrink? $\endgroup$ – curious math guy Dec 15 '20 at 10:24
  • $\begingroup$ @curiousmathguy It's the first section of "Mixed Hodge Structure on the Vanishing Cohomology": math.bgu.ac.il/~kernerdm/eTexts/…. $\endgroup$ – Ben Tighe Dec 15 '20 at 16:52

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