In mathematics, especially algebraic geometry, the decomposition theorem of Beilinson, Bernstein and Deligne or BBD decomposition theorem is a set of results concerning the cohomology of algebraic varieties.
It was originally conjectured by Gelfand and MacPherson.
[1] The first case of the decomposition theorem arises via the hard Lefschetz theorem which gives isomorphisms, for a smooth proper map
of relative dimension d between two projective varieties[2] Here
is the fundamental class of a hyperplane section,
is the n-th derived functor of the direct image.
This derived functor measures the n-th cohomologies of
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.
The decomposition theorem generalizes this fact to the case of a proper, but not necessarily smooth map
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:[3][4] there is an isomorphism in the derived category of sheaves on Y: where
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.
[5] If X is not smooth, then the above results remain true when
is replaced by the intersection cohomology complex
[3] The decomposition theorem was first proved by Beilinson, Bernstein, and Deligne.
[6] 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.
[7] For semismall maps, the decomposition theorem also applies to Chow motives.
from a smooth quasi-projective variety given by
If we set the vanishing locus of
be a proper morphism between complex algebraic varieties such that
that is in an open ball B centered at
Then the restriction map is surjective, where
with the set of regular values of f.[9]