Let $X$ be a complex manifold, $Y\subset X$ a subvariety, and $U:=X\setminus Y$ of codimension $d$. It is well known that the restriction map $$ H^i(X,\mathbb Q)\to H^i(U,\mathbb Q) $$ is an isomorphism for $i<2d-1$. I was wondering if we can replace the sheaf $\mathbb Q_U$ with some constructible sheaf $\mathcal F$ on $U$ in this theorem. Namely, is it true that $$ H^i(X, j_*\mathcal F)\to H^i(U,\mathcal F) $$ is an isomorphism for $i<2d-1$? Here $j\colon U\to X$ is the inclusion map.
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4$\begingroup$ How do you compare the version you proposed and the original theorem? I do not think that $j_*\mathbb Q=\mathbb Q$ (or maybe you are not taking the derived pushforward in $j_*$)? $\endgroup$– Z. MCommented Jun 26 at 15:49
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No. Take $X = \mathbb C^d$, $Y$ a point $P$. Consider a line $L$ through $P$ and let $\mathcal F$ be the pushforward of $\mathbb Q$ from $L \setminus P $ to $U = \mathbb C^d \setminus P$, so that $j_* \mathcal F$ is the pushforward of $\mathbb Q$ from $L$.
Then $H^i(X, j_* \mathcal F)= H^i( L, \mathbb Q)$ vanishes for $i>0$ and is $\mathbb Q$ for $i=0$ while $H^i(U, \mathcal F) = H^i(L\setminus P, Q)$ vanishes for $i>1 $ and is $\mathbb Q$ for $i=0,1$. So in particular they disagree for $i=1$, no matter how large $d$ is.
On the other hand, Poincaré duality implies that for $X$ of dimension $n$ your statement is equivalent to the corestriction map $H^i_c(U, \mathbb Q) \to H^i_c(X,\mathbb Q)$ being an isomorphism for $i> 2n-2d+1$, and this statement does generalize immediately to $H^i_c(U, \mathcal F) \to H^i_c(X,j_* \mathcal F)$ being an isomorphism for constructible $\mathcal F$ on $U$, or even $H^i_c(U,j^* \mathcal F) \to H^i_c(X,\mathcal F)$ being an isomorphism for constructible $\mathcal F$ on $X$, by excision and the cohomological dimension bound for $Y$.
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$\begingroup$ Do you need the constructibility in the last statement? It suffices to control $i^!\mathcal F$, I guess. $\endgroup$– Z. MCommented Jun 26 at 21:15
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$\begingroup$ @Z.M I actually don't see why $i^! \mathcal F$ is relevant - I'm just using excision and the cohomological dimension bound, neither of which should really require constructibility. $\endgroup$ Commented Jun 26 at 21:23
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$\begingroup$ Thanks. I was mistaken. $i^!\mathcal F$ is more relevant to the original question via the fiber sequence $i_*i^!\mathcal F\longrightarrow\mathcal F\longrightarrow j_*j^*\mathcal F$ which gives rise to the fiber sequence $R\Gamma(Y;i^!\mathcal F)\longrightarrow R\Gamma(X;\mathcal F)\longrightarrow R\Gamma(U;\mathcal F)$. As you showed, there is no control for the first term; but If $\mathcal F$ is constructible, is there anything known about the cohomological range with respect to the perverse $t$-structure? $\endgroup$– Z. MCommented Jun 27 at 12:41
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$\begingroup$ @Z.M You're asking about the cohomological range of $i_* i^! \mathcal F$? For $i$ a closed immersion $i_*$ preserves the perverse t-structure so it's equivalent to ask about $i^! \mathcal F$. A constructible sheaf $\mathcal F$ is concentrated in degrees $[-n,0]$ in the perverse $t$ structure for $n = \dim X$ and then $i^!$ has nonnegative cohomological amplitude for the perverse $t$-structure so $i^! \mathcal F$ is concentrated in perverse degree $[-n,\infty)$. $\endgroup$ Commented Jun 27 at 13:27