Lemma 23.2.63ED85C468C237

Proof. For a proper analytic subset $A$ of a connected complex manifold $M$, the restriction map $H^1(M,\mathbb R)\to H^1(M\setminus A,\mathbb R)$ is injective, and it is an isomorphism if $A$ has complex codimension at least two. Indeed, choose a Whitney stratification of $A$. General position moves every one-cycle off the strata, which have real codimension at least two. If those codimensions are at least four, it also moves every two-chain with boundary in $M\setminus A$ off the strata, relative to its boundary. Thus the map $H_1(M\setminus A,\mathbb R)\to H_1(M,\mathbb R)$ is surjective in the first case and bijective in the second, and dualizing over $\mathbb R$ gives the assertions about restriction.

Since $X$ is smooth, there is an analytic subset $A\subset X$ of complex codimension at least two such that $\pi $ is an isomorphism over $U=X\setminus A$. Restriction from $X$ to $U$ is therefore an isomorphism, whereas restriction from $Y$ to $\pi ^{-1}U$ is injective. Every class on $\pi ^{-1}U\simeq U$ extends to $X$ and hence pulls back to a class on $Y$. The commutative diagram of restriction maps now shows that $\pi ^*$ is an isomorphism, and Poincaré duality and the projection formula give the assertion for $\pi _*$. $\square$