# Why is the decomposition theorem awesome?

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.

• what is the theorem like and does it have some classical motivation? – user2438 Dec 9 '09 at 2:47
• 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

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.
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.