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.

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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.