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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. 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 of the decomposition theorem?

Question: What are other examples, especially the "corner" cases?

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 pair ($Y_i$, $\mathcal L_i$) of subvariety together with the local system as $IC_i := j_{!*}\mathcal L_i$.

Now 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 $f_*\mathbb Q = \oplus\\, \mathbb Q[-]$ (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 specific aspects of the decomposition theorem?

Question: What are other examples, especially the "corner" cases?

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 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 "fundamental" cases?

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. 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 of the decomposition theorem?

Question: What are other examples, especially the "corner" cases?

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?

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 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 "fundamental" cases?