Cohomology of resolution of singularity

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.

• No, but it's true for intersection cohomology. – David Hansen Dec 12 '20 at 20:46
• @DavidHansen Great, thank you so much! – curious math guy Dec 12 '20 at 22:04

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 $$\mathbb{C}$$).
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).