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.

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.

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.