Applications for intersection (co)homology and for the Decomposition Theorem for students?

Question

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

Answer 1

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).

Answer 2

A nice example is given by Borho-Macpherson's construction of Weyl group representations (Representations des groupes de Weyl et homologie d'intersection pour les varietes nilpotentes, Comptes rendus. Acad. Sci. Paris, p.707–710 (1981)).

Here's a short description: consider the Springer resolution $p:\tilde{N}\rightarrow N$, where $N\subset \mathfrak{g}$ is the nilpotent cone of some reductive finite dimensional Lie algebra $\mathfrak{g}$ (over $\mathbb{C}$, say). Then, $Rp_{\ast}\mathbb{C}$ carries an action of the Weyl group $W$ of $\mathfrak{g}$, providing a representation of $\mathbb{C}[W]\rightarrow End(Rp_{\ast}\mathbb{C})$; this was an idea of Lusztig, I think. Borho-Macpherson show that this is an isomorphism and the Decomposition Theorem furnishes an identification between irreducible representations of $W$ and irreducible summands of $Rp_{\ast}\mathbb{C}$. For the case $\mathfrak{g}=\mathfrak{gl}_{n}$, this provides a natural bijection between irreducible representations of $S_{n}$ and partitions of $n$; to an irreducible representation $V$ the corresponding summand of $Rp_{\ast}\mathbb{C}$ is supported on a (closure of?) nilpotent orbit associated to some partition.

For $\mathfrak{g}=\mathfrak{gl}_{n}$, you can state Borho-Macpherson's Theorem and give a good idea of the proof with nothing more than some formal properties of IC (fibre squares, for example), the notion of a covering space and some linear algebra. For $\mathfrak{gl}_{n}$, with $n$ small, you can compute examples using some linear algebra; you can also show that the representation corresponding to the subregular nilpotent orbit is the standard irreducible representation $V\subset \mathbb{C}^{n}$, where $\mathbb{C}^{n}$ is the permutation representation (the Springer fibre is a union of $n-1$ projective lines, whose intersection configuration is given by the $A_{n-1}$ Dynkin diagram).

Answer 3

Mikhail,

Coincidentally, I am running a seminar on some of this as well, although I may be assuming a bit more background. But here is an idea which may be suitable for your students. Suppose that $f:X\to Y$ is a projective map from a complex smooth projective variety to a smooth curve. Replace $Y$ by a small disk $D$ centered at $0$. Then $X$ is known to be homotopically equivalent to the fibre $X_0$ over $0$. Thus for $t\in D^*=D-\{0\}$, there is a map $$H^i(X_0)\cong H^i(X)\to H^i(X_t)$$ The image is easily seen to lie in the monodromy invariant part $H^i(X_t)^{\pi_1(D^*)}$. The Local Invariant Cycle theorem shows conversely that, with rational coefficients, any invariant cycle lifts to $H^i(X)$. In the simplest case, where $X_0$ has a single node, this follows from the Picard-Lefschetz formula. But in general the original proof of this theorem, due to Clemens and Schmid, was rather complicated. It made use of the so called limit mixed Hodge structure. There is, however, an easy proof using the decomposition theorem, which is vastly more general (cf. [BBD, cor 6.2.9]). Since the authors don't give a proof, let me sketch a proof of the original version of the LIC theorem:

By the decomposition theorem, $\mathbb{R} f_*\mathbb{Q}$ is a sum of semisimple intersection cohomology sheaves on the original curve $Y$ up to shift. By this, I mean that the summands are $IC(L)[?]$, where $L$ is a local system with semisimple monodromy. Now restrict to $D$ as above. Over $D^\ast$, $f$ is a fibration. By pulling this back to the universal cover $\tilde D^\ast\to D^\ast$, $f$ becomes topologically a product, so we can identify the $L$'s above with $R^if_\ast\mathbb{Q}[-i]$ for $i=0,1,\ldots$. Thus $$ IC(R^if_\ast \mathbb{Q} )[-i] = j_\ast j^\ast R^if_\ast\mathbb{Q}[-i]$$ is a summand of $\mathbb{R} f_\ast\mathbb{Q}$, where $j:D^\ast\to D$ is the inclusion. Therefore $$H^i(X)=H^i(\mathbb{R} f_\ast\mathbb{Q})\to H^i(j_\ast j^\ast R^if_\ast\mathbb{Q}[-i])=H^i(X_t)^{\pi_1(D^\ast)}$$ is surjective.