In mathematics, especially algebraic geometry the decomposition theorem is a set of results concerning the cohomology of algebraic varieties.
Contents
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
Here
This hard Lefschetz isomorphism induces canonical isomorphisms
Moreover, the sheaves
Decomposition for proper maps
The decomposition theorem generalizes this fact to the case of a proper, but not necessarily smooth map
The hard Lefschetz theorem above takes the following form: there is an isomorphism in the derived category of sheaves on Y:
where
Moreover, there is an isomorphism
Finally, the summands at the right hand side are semi-simple perverse sheaves.
If X is not smooth, then the above results remain true when
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.