Easy special cases of the decomposition theorem? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:07:56Z http://mathoverflow.net/feeds/question/37593 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37593/easy-special-cases-of-the-decomposition-theorem Easy special cases of the decomposition theorem? Jan Weidner 2010-09-03T10:06:54Z 2010-09-10T11:18:02Z <p>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.</p> <p>Are there special cases for the decomposition theorem, with "easy" proofs?</p> <p>Are there heuristics, why the decomposition theorem should hold?</p> http://mathoverflow.net/questions/37593/easy-special-cases-of-the-decomposition-theorem/37604#37604 Answer by Donu Arapura for Easy special cases of the decomposition theorem? Donu Arapura 2010-09-03T11:48:02Z 2010-09-10T11:18:02Z <p>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. </p> <blockquote> <p>Theorem. <code>$\mathbb{R} f_*\mathbb{Q}\cong \bigoplus_i R^if_*\mathbb{Q}[-i]$</code>, when $f:X\to Y$ is a smooth projective morphism of varieties over $\mathbb{C}$. (This holds more generally with $\mathbb{Q}_\ell$-coefficients.)</p> <p>Corollary. The Leray spectral sequence degenerates.</p> </blockquote> <p>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. </p> <p>A much less elementary, but more conceptual argument, uses weights. Say $Y$ is smooth and projective, then $E_2^{pq}=H^p(Y, R^qf_*\mathbb{Q})$ <em>should</em> 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. </p> <p>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. </p> <p>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 <a href="http://www.math.purdue.edu/~dvb/seminar.html" rel="nofollow"> here </a> periodically.</p>