The first case of the decomposition theorem arises via the hard Lefschetz theorem which gives isomorphisms, for a smooth proper map of relative dimensiond between two projective varieties Here is the fundamental class of a hyperplane section, is the direct image and is the n-th derived functor of the direct image. This derived functor measures the n-th cohomologies of, for. In fact, the particular case when Y is a point, amounts to the isomorphism This hard Lefschetz isomorphism induces canonical isomorphisms Moreover, the sheaves 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 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 is the total derived functor of and is the i-th truncationwith 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 is replaced by the intersection cohomology complex.
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.
Consider a rational morphism from a smooth quasi-projective variety given by. If we set the vanishing locus of as then there is an induced morphism. We can compute the cohomology of from the intersection cohomology of and subtracting off the cohomology from the blowup along. This can be done using the perverse spectral sequence
