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I saw the statement of the decomposition theorem for perverse sheaves sometime ago. I know that (modulo most of the details) it implies some big theorems in algebraic geometry and gives new proofs for classical important results. I even saw people saying that it is the "deepest theorem in algebraic geometry".

But WHY? Why it is so awesome?

Any answer or comment that helps me to appreciate this theorem will be appreciated. Thanks.

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I saw this article on the Bulletin of AMS website recently; it might be relevant. ams.org/bull/2009-46-04/S0273-0979-09-01260-9/home.html –  Sam Lichtenstein Dec 9 '09 at 2:57
There have been a number of questions on mathoverflow about the decomposition theorem already that you might like to look at. For example, in the side bar of related questions to this one, the first three I see should be of interest. –  Mike Skirvin Dec 9 '09 at 4:07
+1 for asking "Why is X awesome?" –  Scott Morrison Dec 9 '09 at 6:51

3 Answers 3

My answer is maybe more to praise the glory of perverse sheaves than the decomposition theorem exactly, but bear with me. To appreciate the theorem, I'd say first get a sense of what Hodge theory for smooth projective algebraic varieties says: hard Lefschetz etc. (already the fact that the proofs you're likely to see involve harmonic forms and analysis should convince you this is serious stuff). Then try to get a sense of what it means to understand this theorem in families, where things like Hodge filtrations start to appear.

Finally despair of what it might mean to even consider this picture if the "family" you were looking at was just a projective morphism $f\colon Y \to X$, where $Y$ is smooth: the local systems you need for the families version of Hodge theory break down. However, enter perverse sheaves, as sort of singular local systems, and the decomposition theorem says the whole picture is miraculously saved. Viewed this way I think you get a proper sense of how amazing the theorem (and the discovery of perverse sheaves) really is.

P.S. This answer is a poor attempt to convey what others have told me: a better attempt is made in de Cataldo and Migliorini's article

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+1 For "Despair of X" –  Greg Friedman Nov 22 '10 at 5:22

what is the theorem like and does it have some classical motivation?

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This shouldn't be an answer, but a comment to the question above. –  alekzander Dec 9 '09 at 2:49
New users can't leave comments. –  Andrew Critch Dec 9 '09 at 3:06

The article posted in the comment is pretty comprehensive and nice.

The immediate reason this theorem is useful for my research is the proof of the Kazhdan-Lusztig Conjecture. Specifically, using the realization of representations of reductive Lie algebras as modules over (twisted) sheaves of differential operators on a ﬂag variety $G/B$. The Kazhdan-Lusztig conjecture establishes a correspondence between the representations of an algebraic group $G$, to the algebraic-geometric structure of generalized ﬂag varieties $G/B$. In particular, it gives the relation between the characters of Verma modules of Lie algebras and the intersection cohomology on the Schubert varieties.

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