I’ve asked this question on MSE but I don’t get an answer, so I’m trying to ask here.
In the section 3.1 of the paper Intersection forms,topology of maps and motives decomposition for resolution of three folds by de Cataldo and Migliorini:
https://arxiv.org/abs/math/0504554
They prove the following theorem: Let $f: X\to Y$ be a proper birational map of quasi-projective surfaces, $X$ smooth, $Y$ normal. There is a canonical isomorphism:
$Rf_*\mathbb{Q}_X[2]\cong IC_Y\oplus R^2f_*\mathbb{Q}_X[0]$.
And they work locally on $Y$,more precisely, let’s further assume $(Y, p)$ a germ of an analytic normal surface singularity and $f:(X,f^{-1}(p))\to (Y, p)$ a resolution, here $f^{-1}(p)$ is the exceptional divisor.
Under this setting, we can take the stalk at $p$ on both sides, then we have:
$H^1(f^{-1}(P))\cong H^1(IC_Y)$,
at degree $1$. However, consider $j:Y-p\to Y$ the open embedding, since $j^*Rf\mathbb{Q}_X\cong \mathbb{Q}_{Y-p}$ and $IC_Y=\tau_{\leq -1}Rj_*j^*Rf_*\mathbb{Q}_X[2]$, we have:
$H^{-1}(IC_Y)=R^1(j_*\mathbb{Q}_{Y-p})\cong H^1(Y-p)$.
And $H^1(Y-p)$ contain no information about fiber $f^{-1}(p)$, which seems a contradiction. But I don’t know where do I make the mistake, Can somebody help me to find it out? Thanks in advance.
