Which applications of intersection (co)homology and of the (Topological) Decomposition Theorem have most chances to be understood by students?



Applications for intersection (co)homology and for the Decomposition Theorem for students?
I gave a course on the decomposition theorem (and its failure with positive char coefficients) in Bochum. Notes and exercises are here:
<http://people.mpim-bonn.mpg.de/geordie/bochum/>


I found the example of Weierstraß family of elliptic curves quite instructive. One considers the family
$Y^2 = (X-\lambda Z)(X-Z)X$ of elliptic curves over $\mathbb{C}$ (parametrised by $\lambda$).


In a series of exercises (available at the above link) the topology of this example can be pursued in several steps:


1) one tries to understand the local system of $H^1$ away from the singular points 0 and 1. Using ramified covers of $\mathbb{P}^1$ students can calculate the monodromy around the singular points.


(Here one sees the first remarkable fact predicted by the decomposition theorem: even though the monodromy is unipotent around each singularity, the global representation of the free group on two letters is simple. So the decomposition theorem fails in the complex analytic category.)


2) one can calculate the the cohomology of the singular points and sees that it agrees with the invariants of the monodromy.


(Here one sees why the definition of the IC sheaf is "correct" for local systems on $\mathbb{C}$ minus points. This is also an example of the invariant cycle theorem, as Donu points out.)


(More advanced: This also gives an example of a non pointwise pure IC sheaf.)


3) The local systems of $H^0$ and $H^2$ are constant. One can use the hard Lefschetz theorem along the fibres of the map to deduce the splitting of the direct image.


(Here one sees the relative hard Lefschetz theorem in play.)


Other examples that I find really instructive:


1) resolutions of Kleinian surface singularities. (These are also discussed in the notes above.) Here one can connect the decomposition theorem to the non-degeneracy of intersection forms, as in the beautiful work of de Cataldo and Migliorini.


There are notes from a beautiful course Migliorini gave here:


<http://people.mpim-bonn.mpg.de/geordie/Migliorini.pdf>


2) I find the proof of Deligne's theorem on the decomposition theorem for smooth maps quite instructive. Here one really sees "relative Hodge theory" in action: the hard Lefschetz theorem implies the degeneration of the spectral sequence, and to get the semi-simplicity of the local systems of cohomology one needs to use the fact that they are all polarised by an ample line bundle on the fibres, hence admit invariant forms, hence are semi-simple.


3) Another example where the situation is much simpler is the Hilbert scheme of points on a curve, viewed as a resolution of the symmetric power (the Hilbert-Chow morphism).


