Let us consider the following situation: we have a proper map of smooth complex projective algebraic varieties $f:X \to Y$ of dimension $=n$. Let's say we moreover have a decomposition into the derived category of constructible sheaves $$Rf_*\mathbb{Q} \cong \bigoplus_{i \in I}r_{i*}\mathbb{Q}[dim Z_i-n] $$ where $r_i:Z_i \to Y$ is a map finite and birational over a closed subvariety of $Y$.
This actually just a slight more general reformulation of what happens applying the decomposition theorem to the Hilbert scheme of a surface:my question is motivated by https://arxiv.org/pdf/1810.05330.pdf and https://arxiv.org/abs/1012.2583.
Now the isomorphisms $H^{*}(X,\mathbb{Q})\cong H^{*}(Y,Rf_{*}\mathbb{Q})$ and the other one $H^{*}(Y,r_{i*}\mathbb{Q}[dim Z_i-n])=H^{*-n+dim Z_i}(Z_i,\mathbb{Q})$ give us a splitting $$H^{*}(X,\mathbb{Q})=\bigoplus_{i\in I}H^{*-n+dim Z_i}(Z_i,\mathbb{Q}) .$$
I'd like to prove the following statement: the summand $H^{*-n+dim Z_i}(Z_i,\mathbb{Q}) $ into $H^{*}(X,\mathbb{Q}) $ is the image of the following map. Let us call $p:X \times Z_i \to X$ and $q:X \times Z_i \to Z_i$ the projection maps. We denote as $\Gamma_i \subseteq X \times Z_i $ the closed subvariety $X \times_{Y}Z_i$. We then define the map: $$H^{*-n+dim Z_i}(Z_i,\mathbb{Q}) \to H^{*}(X,\mathbb{Q}) $$ $$\alpha \to p_*(q^*\alpha \cup q^*td(Z_i) \cup ch(\mathcal{O}_{\Gamma_i})) .$$
I tried to prove but had no idea how to do that actually. I imagine Grothendick Riemann roch should be involved but do not know how.