# Easy special cases of the decomposition theorem?

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?

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.

• Thanks, for the answer I will accept it. Although if you can add something more in a few months do it :) – Jan Weidner Sep 7 '10 at 6:26
• Should one of the $p$'s in $E*{pq}2=H^p(Y,R^p f∗Q)$ be a $q$', above? – Richard Montgomery Sep 10 '10 at 3:16