There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne.
Here's a refresher: by $IC$ one means the intersection complex, which is just $\mathbb Q$ for a smooth scheme but more complicated for others, and by $IC_i$ one denotes the complex constructed from a subscheme pair ($Y_i$, $Y_i$ \mathcal L_i$) of subvariety together with the local system as $IC_i := j_{!*}\mathcal L_i$.
Now it turns out that for a projective morphism $f: X\to Y$ , turns out 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 $f_*\mathbb Q = IC_Y \oplus F$ (and $F$ should have support on the exceptional divisor.)
- For a smooth algebraic bundle $f_*\mathbb Q = \oplus oplus\, \mathbb Q[-]$ (spectral sequence degenerates)
There are many known applications of the theorem, described, e.g. in the review
The Decomposition Theorem and the topology of algebraic maps* by de Cataldo and Migliorini,
but I wonder if there are more examples that would continue the list above, that is, "corner cases" which highlight particularly one specific aspect aspects of the decomposition theorem?
Question: What are other examples, especially the "corner" cases?
