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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 $E_2^{pq}=H^p(Y, R^pf_*\mathbb{Q})$ 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. 4 added 178 characters in body 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$E_2^{pq}=H^p(Y, R^pf_*\mathbb{Q})$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. 3 added 148 characters in body 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. $\mathbb{R} f_*\mathbb{Q}= f_*\mathbb{Q}\cong \bigoplus_i R^if_*\mathbb{Q}[-i]$, 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$E_2^{pq}=H^p(Y, R^pf_*\mathbb{Q})$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. The If$f$is proper but not smooth, the decomposition theorem also shows that$\mathbb{R} f_*\mathbb{Q}$decomposes into sum of translates of intersection cohomology complexes. This follows from more sophisticated purity considerations, arguments (either in the$\ell\$-adic sense setting as in BBD, or the Hodge theoretic sense setting in Saito's workwork). 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.

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