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I guess this is standard and there is a citable reference. I think the following is an argument which only uses the formalism (e.g. also works in the étale case). (This is an edited version of my first incorrect answer.)

Firstly, if $X \to Y \to Z \stackrel{+1}{\to}$ is a distinguished triangle with $X, Y$ of wt $\ge 0$, then $Z$ is of weight $\ge 0$. (This is easy if one thinks about Frobenius eigenvalues.)

Dually, if $X \to Y \to Z \stackrel{+1}{\to}$ is a dt with $Y, Z$ of wt $\le 0$ then $X$ is of wt $\le 0$.

Now the constant sheaf $k_Y$ on $Y$ is of wt $\le 0$. Hence the dualizing sheaf $\omega_Y$ on $Y$ is of wt $\ge 0$.

Let $f : X \to Y$ be as in your question. Consider the distinguished triangle $K \to f_!f^!\omega_Y \to \omega_Y \stackrel{+1}{\to}$ (where $K$ is defined as the shift of the cone over the adjunction morphism $f_!f^! \to id$). The above remarks show that $K$ is of weights $\ge -1$.

Claim: We are done if we can show $K$ is of weights $\ge 0$.

Proof: Pushing to a point we get a long exact sequence

$\dots \to H^k(Y,K) \to H^k(X,\omega_X) \to H^k(Y,\omega_Y) \to H^{k+1}(Y,K) \to \dots$

now everything in $H^{k+1}(Y,K)$ is of weight $\ge k+1$ (because $*$-pushforward preserves wt $\ge 0$). We conclude that we have a surjection

$gr_W^kH^k(X,\omega_X) \to gr_W^kH^k(X,\omega_Y)$

Finally, because $H_k^{BM}(X) = H^{-k}(X,\omega_X)$ (Borel-Moore homology) the claim follows.

We now give a sketch of how to prove the claim. Because this argument is getting more complicated than I had first intended, I'll give a sketch. I can try to provide more details if it is useful for you.

Consider a weight filtration $W$ on $\omega_Y$ (it is not "the" weight filtration because $\omega_Y$ is not necessarily perverse). I claim that $gr^W_{\le 0}(\omega_Y) = gr^W_0(\omega_Y)$ is a suitable shift-twist of $IC(Y)$. This basic idea is that $\omega_Y$ does not have any sections supported on subvarieties, and any other $IC$ in $gr_{\le 0}(\omega_Y)$ would contribute such a forbidden section.

  Similarly, if $W$ denotes a weight filtration on $X$ then $gr^W_{0}(\omega_X)$ is $IC(X)$ (shift twist).

Now consider the adjunction map $f_!f^!\omega_Y \to \omega_Y$. The weight zero part is given by the map $f_!IC(X) = f_! gr^W_0(\omega_X) \to gr^W_0(\omega_Y)$. Now by the decomposition theorem (here we use surjectivity) $IC(Y)$ occurs as a summand of $f_!IC(X)$ in smallest degree.

Now one deduces (I can only see how to do this using generic smoothness at the moment) that there exists $IC(Y)[?](?) \to f_!\omega_X$ such that the induced map

$$IC(Y)[?](?) \to gr^W_0(\omega_Y) = IC(Y)[?](?)$$

is an isomorphism. We conclude that the triangle

$K \to f_!\omega_X \to \omega_Y \stackrel{+1}{\to}$

can be replaced by a triangle

$K \to L \to gr^W_{\ge 1}(\omega_Y) \stackrel{+1}{\to}$

with $L$ of wts $\ge 0$ and $gr^W_{\ge 1}(\omega_Y)$ of weights $\ge 1$. We conclude that $K$ has weights $\ge 0$ as claimed.

I guess this is standard and there is a citable reference. I think the following is an argument which only uses the formalism (e.g. also works in the étale case). (This is an edited version of my first incorrect answer.)

Firstly, if $X \to Y \to Z \stackrel{+1}{\to}$ is a distinguished triangle with $X, Y$ of wt $\ge 0$, then $Z$ is of weight $\ge 0$. (This is easy if one thinks about Frobenius eigenvalues.)

Dually, if $X \to Y \to Z \stackrel{+1}{\to}$ is a dt with $Y, Z$ of wt $\le 0$ then $X$ is of wt $\le 0$.

Now the constant sheaf $k_Y$ on $Y$ is of wt $\le 0$. Hence the dualizing sheaf $\omega_Y$ on $Y$ is of wt $\ge 0$.

Let $f : X \to Y$ be as in your question. Consider the distinguished triangle $K \to f_!f^!\omega_Y \to \omega_Y \stackrel{+1}{\to}$ (where $K$ is defined as the shift of the cone over the adjunction morphism $f_!f^! \to id$). The above remarks show that $K$ is of weights $\ge -1$.

Claim: We are done if we can show $K$ is of weights $\ge 0$.

Proof: Pushing to a point we get a long exact sequence

$\dots \to H^k(Y,K) \to H^k(X,\omega_X) \to H^k(Y,\omega_Y) \to H^{k+1}(Y,K) \to \dots$

now everything in $H^{k+1}(Y,K)$ is of weight $\ge k+1$ (because $*$-pushforward preserves wt $\ge 0$). We conclude that we have a surjection

$gr_W^kH^k(X,\omega_X) \to gr_W^kH^k(X,\omega_Y)$

Finally, because $H_k^{BM}(X) = H^{-k}(X,\omega_X)$ (Borel-Moore homology) the claim follows.

We now give a sketch of how to prove the claim. Because this argument is getting more complicated than I had first intended, I'll give a sketch. I can try to provide more details if it is useful for you.

Consider a weight filtration $W$ on $\omega_Y$ (it is not "the" weight filtration because $\omega_Y$ is not necessarily perverse). I claim that $gr^W_{\le 0}(\omega_Y) = gr^W_0(\omega_Y)$ is isomorphic to $IC(Y)[d_Y](d_Y)$ (where $d_Y = dim Y$). This basic idea is that $\omega_Y$ does not have any sections supported on subvarieties, and any other $IC$ in $gr_{\le 0}(\omega_Y)$ would contribute such a forbidden section. Similarly, if $W$ denotes a weight filtration on $\omega_X$ then $gr^W_{0}(\omega_X)$ is $IC(X)[d_X](d_X)$.

Now consider the adjunction map $f_!f^!\omega_Y \to \omega_Y$. The weight zero part is given by the map $f_!IC(X) = f_! gr^W_0(\omega_X) \to gr^W_0(\omega_Y)$. Now by the decomposition theorem (here we use surjectivity) $IC(Y)$ occurs as a summand of $f_!IC(X)$ in smallest degree.

Now one deduces (I can only see how to do this using generic smoothness at the moment) that there exists $IC(Y)[?](?) \to f_!\omega_X$ such that the induced map

$$IC(Y)[?](?) \to gr^W_0(\omega_Y) = IC(Y)[?](?)$$

is an isomorphism. We conclude that the triangle

$K \to f_!\omega_X \to \omega_Y \stackrel{+1}{\to}$

can be replaced by a triangle

$K \to L \to gr^W_{\ge 1}(\omega_Y) \stackrel{+1}{\to}$

with $L$ of wts $\ge 0$ and $gr^W_{\ge 1}(\omega_Y)$ of weights $\ge 1$. We conclude that $K$ has weights $\ge 0$ as claimed.

I guess this is standard and there is a citable reference. I think the following is an argument which only uses the formalism (e.g. also works in the étale case).

Firstly, if $X \to Y \to Z \stackrel{+1}{\to}$ is a distinguished triangle with $X, Y$ of wt $\ge 0$, then $Z$ is of weight $\ge 0$. (This is easy if one thinks about Frobenius eigenvalues.)

Dually, if $X \to Y \to Z \stackrel{+1}{\to}$ is a dt with $Y, Z$ of wt $\le 0$ then $X$ is of wt $\le 0$.

Now the constant sheaf $k_Y$ on $Y$ is of wt $\le 0$. Hence the dualizing sheaf $\omega_Y$ on $Y$ is of wt $\ge 0$.

Let $f : X \to Y$ be as in your question. Consider the distinguished triangle $K \to f_!f^!\omega_Y \to \omega_Y \stackrel{+1}{\to}$ (where $K$ is defined as the shift of the cone over the adjunction morphism $f_!f^! \to id$). The above remarks imply that $K$ is of weight $\ge 0$ (as $f^!\omega_Y = \omega_X$ and $f_! = f_*$ preserves wts $\ge 0$).

Pushing to a point we get a long exact sequence

$\dots \to H^k(Y,K) \to H^k(X,\omega_X) \to H^k(Y,\omega_Y) \to H^{k+1}(Y,K) \to \dots$

now everything in $H^{k+1}(Y,K)$ is of weight $\ge k+1$ (because $*$-pushforward preserves wt $\ge 0$). We conclude that we have a surjection

$gr_W^kH^k(X,\omega_X) \to gr_W^kH^k(X,\omega_Y)$

Finally, because $H_k^{BM}(X) = H^{-k}(X,\omega_X)$ (Borel-Moore homology) the result follows.

I guess this is standard and there is a citable reference. I think the following is an argument which only uses the formalism (e.g. also works in the étale case). (This is an edited version of my first incorrect answer.)

Firstly, if $X \to Y \to Z \stackrel{+1}{\to}$ is a distinguished triangle with $X, Y$ of wt $\ge 0$, then $Z$ is of weight $\ge 0$. (This is easy if one thinks about Frobenius eigenvalues.)

Dually, if $X \to Y \to Z \stackrel{+1}{\to}$ is a dt with $Y, Z$ of wt $\le 0$ then $X$ is of wt $\le 0$.

Now the constant sheaf $k_Y$ on $Y$ is of wt $\le 0$. Hence the dualizing sheaf $\omega_Y$ on $Y$ is of wt $\ge 0$.

Let $f : X \to Y$ be as in your question. Consider the distinguished triangle $K \to f_!f^!\omega_Y \to \omega_Y \stackrel{+1}{\to}$ (where $K$ is defined as the shift of the cone over the adjunction morphism $f_!f^! \to id$). The above remarks show that $K$ is of weights $\ge -1$.

Claim: We are done if we can show $K$ is of weights $\ge 0$.

Proof: Pushing to a point we get a long exact sequence

$\dots \to H^k(Y,K) \to H^k(X,\omega_X) \to H^k(Y,\omega_Y) \to H^{k+1}(Y,K) \to \dots$

now everything in $H^{k+1}(Y,K)$ is of weight $\ge k+1$ (because $*$-pushforward preserves wt $\ge 0$). We conclude that we have a surjection

$gr_W^kH^k(X,\omega_X) \to gr_W^kH^k(X,\omega_Y)$

Finally, because $H_k^{BM}(X) = H^{-k}(X,\omega_X)$ (Borel-Moore homology) the claim follows.

We now give a sketch of how to prove the claim. Because this argument is getting more complicated than I had first intended, I'll give a sketch. I can try to provide more details if it is useful for you.

Consider a weight filtration $W$ on $\omega_Y$ (it is not "the" weight filtration because $\omega_Y$ is not necessarily perverse). I claim that $gr^W_{\le 0}(\omega_Y) = gr^W_0(\omega_Y)$ is a suitable shift-twist of $IC(Y)$. This basic idea is that $\omega_Y$ does not have any sections supported on subvarieties, and any other $IC$ in $gr_{\le 0}(\omega_Y)$ would contribute such a forbidden section.

Similarly, if $W$ denotes a weight filtration on $X$ then $gr^W_{0}(\omega_X)$ is $IC(X)$ (shift twist).

Now consider the adjunction map $f_!f^!\omega_Y \to \omega_Y$. The weight zero part is given by the map $f_!IC(X) = f_! gr^W_0(\omega_X) \to gr^W_0(\omega_Y)$. Now by the decomposition theorem (here we use surjectivity) $IC(Y)$ occurs as a summand of $f_!IC(X)$ in smallest degree.

Now one deduces (I can only see how to do this using generic smoothness at the moment) that there exists $IC(Y)[?](?) \to f_!\omega_X$ such that the induced map

$$IC(Y)[?](?) \to gr^W_0(\omega_Y) = IC(Y)[?](?)$$

is an isomorphism. We conclude that the triangle

$K \to f_!\omega_X \to \omega_Y \stackrel{+1}{\to}$

can be replaced by a triangle

$K \to L \to gr^W_{\ge 1}(\omega_Y) \stackrel{+1}{\to}$

with $L$ of wts $\ge 0$ and $gr^W_{\ge 1}(\omega_Y)$ of weights $\ge 1$. We conclude that $K$ has weights $\ge 0$ as claimed.

I guess this is standard and there is a citable reference. I think the following is an argument which only uses the formalism (e.g. also works in the étale case).

Firstly, if $X \to Y \to Z \stackrel{+1}{\to}$ is a distinguished triangle with $X, Y$ of wt $\ge 0$, then $Z$ is of weight $\ge 0$. (This is easy if one thinks about Frobenius eigenvalues.)

Dually, if $X \to Y \to Z \stackrel{+1}{\to}$ is a dt with $Y, Z$ of wt $\le 0$ then $X$ is of wt $\le 0$.

Now the constant sheaf $k_Y$ on $Y$ is of wt $\le 0$. Hence the dualizing sheaf $\omega_Y$ on $Y$ is of wt $\ge 0$.

Let $f : X \to Y$ be as in your question. Consider the distinguished triangle $K \to f_!f^!\omega_Y \to \omega_Y \stackrel{+1}{\to}$ (where $K$ is defined as the shift of the cone over the adjunction morphism $f_!f^! \to id$). The above remarks imply that $K$ is of weight $\ge 0$ (as $f^!\omega_Y = \omega_X$ and $f_! = f_*$ preserves wts $\ge 0$).

Pushing to a point we get a long exact sequence

$\dots \to H^k(Y,K) \to H^k(X,\omega_X) \to H^k(X,\omega_Y) \to H^{k+1}(Y,K) \to \dots$

now everything in $H^{k+1}(Y,K)$ is of weight $\ge k+1$ (because $*$-pushforward preserves wt $\ge 0$). We conclude that we have a surjection

$gr_W^kH^k(X,\omega_X) \to gr_W^kH^k(X,\omega_Y)$

Finally, because $H_k^{BM}(X) = H^{-k}(X,\omega_X)$ (Borel-Moore homology) the result follows.

I guess this is standard and there is a citable reference. I think the following is an argument which only uses the formalism (e.g. also works in the étale case).

Firstly, if $X \to Y \to Z \stackrel{+1}{\to}$ is a distinguished triangle with $X, Y$ of wt $\ge 0$, then $Z$ is of weight $\ge 0$. (This is easy if one thinks about Frobenius eigenvalues.)

Dually, if $X \to Y \to Z \stackrel{+1}{\to}$ is a dt with $Y, Z$ of wt $\le 0$ then $X$ is of wt $\le 0$.

Now the constant sheaf $k_Y$ on $Y$ is of wt $\le 0$. Hence the dualizing sheaf $\omega_Y$ on $Y$ is of wt $\ge 0$.

Let $f : X \to Y$ be as in your question. Consider the distinguished triangle $K \to f_!f^!\omega_Y \to \omega_Y \stackrel{+1}{\to}$ (where $K$ is defined as the shift of the cone over the adjunction morphism $f_!f^! \to id$). The above remarks imply that $K$ is of weight $\ge 0$ (as $f^!\omega_Y = \omega_X$ and $f_! = f_*$ preserves wts $\ge 0$).

Pushing to a point we get a long exact sequence

$\dots \to H^k(Y,K) \to H^k(X,\omega_X) \to H^k(Y,\omega_Y) \to H^{k+1}(Y,K) \to \dots$

now everything in $H^{k+1}(Y,K)$ is of weight $\ge k+1$ (because $*$-pushforward preserves wt $\ge 0$). We conclude that we have a surjection

$gr_W^kH^k(X,\omega_X) \to gr_W^kH^k(X,\omega_Y)$

Finally, because $H_k^{BM}(X) = H^{-k}(X,\omega_X)$ (Borel-Moore homology) the result follows.