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 direct image (pushforward) and
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.
If we set the vanishing locus of
and subtracting off the cohomology from the blowup along
This can be done using the perverse spectral sequence Let
be a proper morphism between complex algebraic varieties such that
is the fundamental group of the intersection of