Decomposition theorem of Beilinson, Bernstein and Deligne

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, Deligne and Gabber. 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.{{citation|author1-link=Mark de Cataldo == 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. ==References==