Let $f:X\to Y$ be a proper surjection of complex algebraic varieties. Let $H_i$ denote Borel-Moore homology. Then $$ \mathrm{Gr}^W_{-k} H_k(X) \to \mathrm{Gr}^W_{-k} H_k(Y) $$ is surjective.

Question: Does anyone know a reference for this fact?

I have a proof, but it's not as simple as it could be. And it uses generic smoothness of $f$, so it's only valid in characteristic zero.

The cycle class map from Chow groups to Borel-Moore homology lands in the lowest weight part, so the above is somehow analogous to the fact that proper pushforward is surjective in Chow for a surjective map.

If $f$ instead is an open immersion, then the induced map on lowest weights is also surjective. This is similarly analogous to the fact that flat pullback is surjective in Chow for open immersions. But here the proof for the result in Borel-Moore homology is very simple, which is one reason I think there should be a simple proof of the above result, too.

Addendum. Here's my proof, it's very similar to the one linked to by novice. The proof in Lewis's book doesn't need generic smoothness, but it assumes instead that $f$ has a multisection.

Note first that the if $f$ is in addition smooth, then $H_k(X) \to H_k(Y)$ is onto and there's not much to it. For the general case take $U \subset X$ where $f$ is smooth, and let $Z = X \setminus U$. Then there is a map between the long exact sequences $$ \cdots \to H_k(Z) \to H_k(X) \to H_k(U) \to H_{k-1}(Z) \to \cdots $$ and $$ \cdots \to H_k(f(Z)) \to H_k(Y) \to H_k(Y \setminus f(Z)) \to H_{k-1}(f(Z)) \to \cdots $$ as follows. The maps $H_\bullet(X) \to H_\bullet(Y)$ and $H_\bullet(Z) \to H_\bullet(f(Z))$ are the obvious ones. The map $H_\bullet(U) \to H_\bullet(Y \setminus f(Z))$ is the composite $$ H_\bullet(U) \to H_\bullet(X \setminus f^{-1}(f(Z))) \to H_\bullet(Y \setminus f(Z)). $$

Now apply $W_{-k}$ to the long exact sequences. The two maps $H_k(Z) \to H_k(f(Z))$ and $H_k(U) \to H_k(Y \setminus f(Z))$ are surjective on lowest weights: the former by noetherian induction and the latter because it's the composition of pullback for an open immersion and a pushforward for a smooth proper morphism. Since in addition $W_{-k}H_{k-1}(Z) = W_{-k}H_{k-1}(f(Z)) = 0$ the result follows by the four lemma.

Let $f:X\to Y$ be a proper surjection of complex algebraic varieties. Let $H_i$ denote Borel-Moore homology. Then $$ \mathrm{Gr}^W_{-k} H_k(X) \to \mathrm{Gr}^W_{-k} H_k(Y) $$ is surjective.

Question: Does anyone know a reference for this fact?

I have a proof, but it's not as simple as it could be. And it uses generic smoothness of $f$, so it's only valid in characteristic zero.

The cycle class map from Chow groups to Borel-Moore homology lands in the lowest weight part, so the above is somehow analogous to the fact that proper pushforward is surjective in Chow for a surjective map.

If $f$ instead is an open immersion, then the induced map on lowest weights is also surjective. This is similarly analogous to the fact that flat pullback is surjective in Chow for open immersions. But here the proof for the result in Borel-Moore homology is very simple, which is one reason I think there should be a simple proof of the above result, too.

Addendum. Here's my proof, it's very similar to the one linked to by novice. However, the proof in Lewis's book doesn't need generic smoothness; it uses instead the existence of a subvariety mapping generically finitely onto $Y$, as in the suggestion of ACL below.

Note first that the if $f$ is in addition smooth, then $H_k(X) \to H_k(Y)$ is onto and there's not much to it. For the general case take $U \subset X$ where $f$ is smooth, and let $Z = X \setminus U$. Then there is a map between the long exact sequences $$ \cdots \to H_k(Z) \to H_k(X) \to H_k(U) \to H_{k-1}(Z) \to \cdots $$ and $$ \cdots \to H_k(f(Z)) \to H_k(Y) \to H_k(Y \setminus f(Z)) \to H_{k-1}(f(Z)) \to \cdots $$ as follows. The maps $H_\bullet(X) \to H_\bullet(Y)$ and $H_\bullet(Z) \to H_\bullet(f(Z))$ are the obvious ones. The map $H_\bullet(U) \to H_\bullet(Y \setminus f(Z))$ is the composite $$ H_\bullet(U) \to H_\bullet(X \setminus f^{-1}(f(Z))) \to H_\bullet(Y \setminus f(Z)). $$

Now apply $W_{-k}$ to the long exact sequences. The two maps $H_k(Z) \to H_k(f(Z))$ and $H_k(U) \to H_k(Y \setminus f(Z))$ are surjective on lowest weights: the former by noetherian induction and the latter because it's the composition of pullback for an open immersion and a pushforward for a smooth proper morphism. Since in addition $W_{-k}H_{k-1}(Z) = W_{-k}H_{k-1}(f(Z)) = 0$ the result follows by the four lemma.

Let $f:X\to Y$ be a proper surjection of complex algebraic varieties. Let $H_i$ denote Borel-Moore homology. Then $$ Gr^W_{-k} H^k(X) \to Gr^W_{-k} H^k(Y) $$ is surjective.

Question: Does anyone know a reference for this fact?

I have a proof, but it's not as simple as it could be. And it uses generic smoothness of $f$, so it's only valid in characteristic zero.

The cycle class map from Chow groups to Borel-Moore homology lands in the lowest weight part, so the above is somehow analogous to the fact that proper pushforward is surjective in Chow for a surjective map.

If $f$ instead is an open immersion, then the induced map on lowest weights is also surjective. This is similarly analogous to the fact that flat pullback is surjective in Chow for open immersions. But here the proof for the result in Borel-Moore homology is very simple, which is one reason I think there should be a simple proof of the above result, too.

Let $f:X\to Y$ be a proper surjection of complex algebraic varieties. Let $H_i$ denote Borel-Moore homology. Then $$ \mathrm{Gr}^W_{-k} H_k(X) \to \mathrm{Gr}^W_{-k} H_k(Y) $$ is surjective.

Question: Does anyone know a reference for this fact?

I have a proof, but it's not as simple as it could be. And it uses generic smoothness of $f$, so it's only valid in characteristic zero.

The cycle class map from Chow groups to Borel-Moore homology lands in the lowest weight part, so the above is somehow analogous to the fact that proper pushforward is surjective in Chow for a surjective map.

If $f$ instead is an open immersion, then the induced map on lowest weights is also surjective. This is similarly analogous to the fact that flat pullback is surjective in Chow for open immersions. But here the proof for the result in Borel-Moore homology is very simple, which is one reason I think there should be a simple proof of the above result, too.

Addendum. Here's my proof, it's very similar to the one linked to by novice. The proof in Lewis's book doesn't need generic smoothness, but it assumes instead that $f$ has a multisection.

Note first that the if $f$ is in addition smooth, then $H_k(X) \to H_k(Y)$ is onto and there's not much to it. For the general case take $U \subset X$ where $f$ is smooth, and let $Z = X \setminus U$. Then there is a map between the long exact sequences $$ \cdots \to H_k(Z) \to H_k(X) \to H_k(U) \to H_{k-1}(Z) \to \cdots $$ and $$ \cdots \to H_k(f(Z)) \to H_k(Y) \to H_k(Y \setminus f(Z)) \to H_{k-1}(f(Z)) \to \cdots $$ as follows. The maps $H_\bullet(X) \to H_\bullet(Y)$ and $H_\bullet(Z) \to H_\bullet(f(Z))$ are the obvious ones. The map $H_\bullet(U) \to H_\bullet(Y \setminus f(Z))$ is the composite $$ H_\bullet(U) \to H_\bullet(X \setminus f^{-1}(f(Z))) \to H_\bullet(Y \setminus f(Z)). $$

Now apply $W_{-k}$ to the long exact sequences. The two maps $H_k(Z) \to H_k(f(Z))$ and $H_k(U) \to H_k(Y \setminus f(Z))$ are surjective on lowest weights: the former by noetherian induction and the latter because it's the composition of pullback for an open immersion and a pushforward for a smooth proper morphism. Since in addition $W_{-k}H_{k-1}(Z) = W_{-k}H_{k-1}(f(Z)) = 0$ the result follows by the four lemma.