Decomposition theorem

In mathematics, especially algebraic geometry the decomposition theorem is a set of results concerning the cohomology of algebraic varieties.
Statement.
Decomposition for smooth proper maps.
The first case of the decomposition theorem arises via the hard Lefschetz theorem which gives isomorphisms, for a smooth proper map formula_1 of relative dimension "d" between two projective varieties
Here formula_3 is the fundamental class of a hyperplane section, formula_4 is the direct image (pushforward) and formula_5 is the "n"-th derived functor of the direct image. This derived functor measures the "n"-th cohomologies of formula_6, for formula_7.
In fact, the particular case when "Y" is a point, amounts to the isomorphism
This hard Lefschetz isomorphism induces canonical isomorphisms
Moreover, the sheaves formula_10 appearing in this decomposition are local systems, i.e., locally free sheaves of Q-vector spaces, which are moreover semisimple, i.e., a direct sum of local systems without nontrivial local subsystems.
Decomposition for proper maps.
The decomposition theorem generalizes this fact to the case of a proper, but not necessarily smooth map formula_1 between varieties. In a nutshell, the results above remain true when the notion of local systems is replaced by perverse sheaves.
The hard Lefschetz theorem above takes the following form: there is an isomorphism in the derived category of sheaves on "Y":
where formula_13 is the total derived functor of formula_4 and formula_15 is the "i"-th truncation with respect to the "perverse" t-structure.
Moreover, there is an isomorphism
where the summands are semi-simple perverse-sheaves, meaning they are direct sums of push-forwards of intersection cohomology sheaves.
If "X" is not smooth, then the above results remain true when formula_17 is replaced by the intersection cohomology complex formula_18.
Proofs.
The decomposition theorem was first proved by Beilinson, Bernstein, and Deligne. Their proof is based on the usage of weights on l-adic sheaves in positive characteristic. A different proof using mixed Hodge modules was given by Saito. A more geometric proof, based on the notion of semismall maps was given by de Cataldo and Migliorini.
For semismall maps, the decomposition theorem also applies to Chow motives.
Applications of the Decomposition Theorem.
Cohomology of a Rational Lefschetz Pencil.
Consider a rational morphism formula_19 from a smooth quasi-projective variety given by formula_20. If we set the vanishing locus of formula_21 as formula_22 then there is an induced morphism formula_23. We can compute the cohomology of formula_24 from the intersection cohomology of formula_25 and subtracting off the cohomology from the blowup along formula_22. This can be done using the perverse spectral sequence