There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne.

Here's a refresher: by $IC$ I mean the intersection complex, the one that is just $\mathbb Q$ for a smooth scheme but more complicated for others, and by $IC_i$ its version correctly extended (the $j_{!*}$ notation) from a subscheme of $Y_i$ (with bundle $\mathcal L_i$).

Now it turns out that for a morphism $f: X\to Y$, you can decompose in the derived category $$f_*IC = \oplus IC_i[n_i].$$ The special beauty of this decomposition theorem is in its examples. Here are some I think I know:

• For a free action of a group G on some X, you get the decomposition by representation of G.
• For a resolution of singularities, you get $f_*\mathbb Q = \mathbb Q \oplus F$, ($F$ has support on the exceptional divisor.)
• For a smooth algebraic bundle $f_*\mathbb Q = \oplus \mathbb Q[-]$ (spectral sequence degenerates)

Question: What are other examples, especially the important special cases?