# Decomposition theorem of Beilinson, Bernstein and Deligne

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.

## 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 $f:X\to Y$ of relative dimension d between two projective varieties
    $-\cup \eta ^{i}:R^{d-i}f_{*}(\mathbb {Q} ){\stackrel {\cong }{\to }}R^{d+i}f_{*}(\mathbb {Q} ).$
Here $\eta$ is the fundamental class of a hyperplane section, $f_{*}$ is the direct image (pushforward) and $R^{n}f_{*}$ is the n-th derived functor of the direct image. This derived functor measures the n-th cohomologies of $f^{-1}(U)$, for $U\subset Y$. In fact, the particular case when Y is a point, amounts to the isomorphism
    $-\cup \eta ^{i}:H^{d-i}(X,\mathbb {Q} ){\stackrel {\cong }{\to }}H^{d+i}(X,\mathbb {Q} ).$
This hard Lefschetz isomorphism induces canonical isomorphisms
    $Rf_{*}(\mathbb {Q} ){\stackrel {\cong }{\to }}\bigoplus _{i=-d}^{d}R^{d+i}f_{*}(\mathbb {Q} )[-d-i].$
Moreover, the sheaves $R^{d+i}f_{*}\mathbb {Q}$ 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 $f:X\to Y$ 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:
    $^{p}H^{-i}(Rf_{*}\mathbb {Q} )\cong^{p}H^{+i}(Rf_{*}\mathbb {Q} ),$
where $Rf_{*}$ is the total derived functor of $f_{*}$ and $^{p}H^{i}$ is the i-th truncation with respect to the perverse t-structure.
Moreover, there is an isomorphism
    $Rf_{*}IC_{X}^{\bullet }\cong \bigoplus _{i}^{p}H^{i}(Rf_{*}IC_{X}^{\bullet })[-i].$
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 $\mathbb {Q} [\dim X]$ is replaced by the intersection cohomology complex $IC$.

## 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 theorem
### Cohomology of a Rational Lefschetz Pencil

Consider a rational morphism $f:X\rightarrow \mathbb {P} ^{1}$ from a smooth quasi-projective variety given by $[f_{1}(x):f_{2}(x)]$. If we set the vanishing locus of $f_{1},f_{2}$ as $Y$ then there is an induced morphism ${\tilde {X}}=Bl_{Y}(X)\to \mathbb {P} ^{1}$. We can compute the cohomology of $X$ from the intersection cohomology of $Bl_{Y}(X)$ and subtracting off the cohomology from the blowup along $Y$. This can be done using the perverse spectral sequence
    $E_{2}^{l,m}=H^{l}(\mathbb {P} ^{1};^{\mathfrak {p}}{\mathcal {H}}^{m}(IC_{\tilde {X}}^{\bullet }(\mathbb {Q} ))\Rightarrow IH^{l+m}({\tilde {X}};\mathbb {Q} )\cong H^{l+m}(X;\mathbb {Q} )$

### Local invariant cycle theorem

Let $f:X\to Y$ be a proper morphism between complex algebraic varieties such that $X$ is smooth. Also, let $y_{0}$ be a regular value of $f$ that is in an open ball B centered at $y$. Then the restriction map
    $\operatorname {H} ^{*}(f^{-1}(y),\mathbb {Q} )=\operatorname {H} ^{*}(f^{-1}(B),\mathbb {Q} )\to \operatorname {H} ^{*}(f^{-1}(y_{0}),\mathbb {Q} )^{\pi _{1,{\textrm {loc}}}}$
is surjective, where $\pi _{1,{\textrm {loc}}}$ is the fundamental group of the intersection of $B$ with the set of regular values of f.