Easy special cases of the decomposition theorem?

Question

The decomposition theorem states roughly, that the pushforward of an IC complex, along a proper map decomposes into a direct sum of shifted IC complexes.

Are there special cases for the decomposition theorem, with "easy" proofs?

Are there heuristics, why the decomposition theorem should hold?

Answer 1

Well, it depends on what you mean by "easy". A special case, which I find very instructive, is a theorem of Deligne from the late 1960's.

Theorem. $\mathbb{R} f_*\mathbb{Q}\cong \bigoplus_i R^if_*\mathbb{Q}[-i]$, when $f:X\to Y$ is a smooth projective morphism of varieties over $\mathbb{C}$. (This holds more generally with $\mathbb{Q}_\ell$-coefficients.)

Corollary. The Leray spectral sequence degenerates.

The result was deduced from the hard Lefschetz theorem. An outline of a proof (of the corollary) can be found in Griffiths and Harris. It is tricky but essentially elementary.

A much less elementary, but more conceptual argument, uses weights. Say $Y$ is smooth and projective, then $E_2^{pq}=H^p(Y, R^qf_*\mathbb{Q})$ should be pure of weight $p+q$ (in the sense of Hodge theory or $\ell$-adic cohomology). Since $$d_2: E_2^{pq}\to E_2^{p+2,q-1}$$ maps a structure of one weight to another it must vanish. Similarly for higher differentials.

If $f$ is proper but not smooth, the decomposition theorem shows that $\mathbb{R} f_*\mathbb{Q}$ decomposes into sum of translates of intersection cohomology complexes. This follows from more sophisticated purity arguments (either in the $\ell$-adic setting as in BBD, or the Hodge theoretic setting in Saito's work). There is also a newer proof due to de Cataldo and Migliorini which seems a bit more geometric.

I have been working through some of this stuff slowly. So I may have more to say in a few months time. Rather than updating this post, it may be more efficient for the people interested to check here periodically.