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firstly, I would know if my very basic intuition on perverse sheaves is correct .

secondly, I would have some clarification in what perverse sheaves behaves better than regular sheaves .

my intuition is :

In smooth cases perverse t-structures is approximately the same as ordinary t-structures (up to translation by the dimension of the space) But in the singular case the perverse sheaves behave much better than the ordinary sheaves so one can actually imagine that the perverse sheaves are the correct category, it is a more correct category than the category of ordinary sheaves and the reason for this is that the construction of the perverse sheaves is actually the gluing of the ordinary t structures on all strata.

The intuition for this construction is, a stratification of a space is supposed to divide the space into smooth elements which makes it possible to “correct the singularity” and therefore it makes sense to think that the correct object is perverse sheaves and not ordinary sheaves In addition it is possible to prove that the perverse sheaves are stacks which means that they behave just like sheaves.

I would be grateful is someone could said me if this is a correct explanation\intuition(and by the way my intuition on stratification is correct?) ?

and if someone have another intuition on perverse sheaves I am absolutely open for other intuition !

thanks in advance !!

Tim Campion
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Amos Kaminski
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    A couple of points. 1) When $X$ is smooth, a locally constant sheaf is perverse up to translation. But the converse isn't true (expect in trivial cases). 2) If you understand a bit about $D$-modules, then on the $D$-module side, under Riemann-Hilbert, perverse sheaves look very natural. – Donu Arapura Aug 28 '20 at 13:43
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    "In smooth cases perverse t-structures is approximately the same as ordinary t-structures" - This depends on what you mean by "smooth cases". For example, the perverse t-structure for (algebraically) constructible complexes on $\mathbb A^1 = \mathbb C$ is interesting - not just a shift of the usual t-structure - even though all the strata and links are themselves smooth. E.g. If $j:\mathbb C^\times \hookrightarrow \mathbb C$ is the open inclusion, $Rj_\ast \mathbb{Q}_{\mathbb{C}^\times}[1]$ is perverse, but it has ordinary cohomology in degrees -1 and 0. – Sam Gunningham Aug 29 '20 at 08:37
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    When I said in the smooth cases I am meaning smooth sheaves on essentially smooth and equidimensional. scheme ... – Amos Kaminski Aug 29 '20 at 09:10
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    @Anonyme I've edited to make sure that you have at least one top-level tag on your question. This is considered good practice and is also a good way to maximize the number of relevant users who see your question. Since you already had 5 tags (the maximum number), I had to delete one (in this case "t-structures", which is only watched by 3 people). Feel free to re-adjust the tags as you see fit, but I do recommend having at least one top-level tag. – Tim Campion Aug 30 '20 at 20:16
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    My current favorite reference is "The Decomposition Theorem and the Topology of Algebraic Maps". A very nice example if you know a little toric geometry is the section on the application of the decomposition theorem to toric varieties. In toric geometry, everything in sight consists of T-orbits and closures of T-orbits. For a very down to earth example of cases in which intersection homology works more nicely than regular homology or cohomology is that intersection is a purely combinatorial invariant of the polytope from which you construct the variety, unlike regular cohomology. – Marc Besson Aug 31 '20 at 03:18
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    One difficulty with this question is that there is no way to give a short answer. So I will second Marc Besson's suggestion. Here is a link to the article https://arxiv.org/abs/0712.0349 – Donu Arapura Aug 31 '20 at 14:49
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    If you are looking for a more topological intuition, I would recommend MacPherson's lecture notes, which were mentioned in this other question. – Aurelio Aug 31 '20 at 14:57

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