7 reworded a bit

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 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? 6 worked up 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$Y_i$together with the local system as$j_{!*}\mathcal L_i$. 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$f_*\mathbb Q = IC_Y \oplus F$(and$F$should have support on the exceptional divisor.) • For a smooth algebraic bundle$f_*\mathbb Q = \oplus \mathbb Q[-]$(spectral sequence degenerates) There are many known applications of the theorem, described, e.g. this in the review but I wonder if there are more important examples that would continue the list above, that is, "corner cases" which highlight particularly one specific aspect of the decomposition theorem, as do the examples above? Question: What are other examples, especially the "fundamental" corner" cases? 5 a bi There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne. Here's a refresher: by$IC$I mean one means the intersection complex, the one that which is just$\mathbb Q$for a smooth scheme but more complicated for others, and by$IC_i$its version correctly extended (one denotes the$j_{!*}$notation) complex constructed from a subscheme of$Y_i$(together with bundle the local system as$\mathcal L_i$)j_{!*}\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 $f_*\mathbb Q = \mathbb Q IC_Y \oplus F$ , ($F$ has and $F$ should have support on the exceptional divisor.)
• For a smooth algebraic bundle $f_*\mathbb Q = \oplus \mathbb Q[-]$ (spectral sequence degenerates)

There are many known applications of the theorem, described, e.g. this review, but I wonder if there are more important "corner cases" which highlight particularly one aspect of the decomposition theorem, as do the examples above?

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

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