Decomposition theorem of Beilinson, Bernstein and Deligne

In mathematics, especially algebraic geometry, the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber or BBDG 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 ${\displaystyle f:X\to Y}$ of relative dimension d between two projective varieties²

${\displaystyle -\cup \eta ^{i}:R^{d-i}f_{*}(\mathbb {Q} ){\stackrel {\cong }{\to }}R^{d+i}f_{*}(\mathbb {Q} ).}$

Here ${\displaystyle \eta }$ is the fundamental class of a hyperplane section, ${\displaystyle f_{*}}$ is the direct image (pushforward) and ${\displaystyle R^{n}f_{*}}$ is the n-th derived functor of the direct image. This derived functor measures the n-th cohomologies of ${\displaystyle f^{-1}(U)}$, for ${\displaystyle U\subset Y}$. In fact, the particular case when Y is a point, amounts to the isomorphism

${\displaystyle -\cup \eta ^{i}:H^{d-i}(X,\mathbb {Q} ){\stackrel {\cong }{\to }}H^{d+i}(X,\mathbb {Q} ).}$

This hard Lefschetz isomorphism induces canonical isomorphisms

${\displaystyle Rf_{*}(\mathbb {Q} ){\stackrel {\cong }{\to }}\bigoplus _{i=-d}^{d}R^{d+i}f_{*}(\mathbb {Q} )[-d-i].}$

Moreover, the sheaves ${\displaystyle R^{d+i}f_{*}\mathbb {Q} }$ 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.

The decomposition theorem generalizes this fact to the case of a proper, but not necessarily smooth map ${\displaystyle f:X\to Y}$ 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:³⁴ there is an isomorphism in the derived category of sheaves on Y:

${\displaystyle {}^{p}H^{-i}(Rf_{*}\mathbb {Q} )\cong {}^{p}H^{+i}(Rf_{*}\mathbb {Q} ),}$

where ${\displaystyle Rf_{*}}$ is the total derived functor of ${\displaystyle f_{*}}$ and ${\displaystyle {}^{p}H^{i}}$ is the i-th truncation with respect to the perverse t-structure.

Moreover, there is an isomorphism

${\displaystyle Rf_{*}IC_{X}^{\bullet }\cong \bigoplus _{i}{}^{p}H^{i}(Rf_{*}IC_{X}^{\bullet })[-i].}$

where the summands are semi-simple perverse-sheaves, meaning they are direct sums of push-forwards of intersection cohomology sheaves.⁵

If X is not smooth, then the above results remain true when ${\displaystyle \mathbb {Q} [\dim X]}$ is replaced by the intersection cohomology complex ${\displaystyle IC}$.³

The decomposition theorem was first proved by Beilinson, Bernstein, Deligne and Gabber.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.⁸

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

${\displaystyle E_{2}^{l,m}=H^{l}(\mathbb {P} ^{1};{}^{\mathfrak {p}}{\mathcal {H}}^{m}(IC_{\tilde {X}}^{\bullet }(\mathbb {Q} ))\Rightarrow IH^{l+m}({\tilde {X}};\mathbb {Q} )\cong H^{l+m}(X;\mathbb {Q} )}$

Let ${\displaystyle f:X\to Y}$ be a proper morphism between complex algebraic varieties such that ${\displaystyle X}$ is smooth. Also, let ${\displaystyle y_{0}}$ be a regular value of ${\displaystyle f}$ that is in an open ball B centered at ${\displaystyle y}$. Then the restriction map

${\displaystyle \operatorname {H} ^{*}(f^{-1}(y),\mathbb {Q} )=\operatorname {H} ^{*}(f^{-1}(B),\mathbb {Q} )\to \operatorname {H} ^{*}(f^{-1}(y_{0}),\mathbb {Q} )^{\pi _{1,{\textrm {loc}}}}}$

is surjective, where ${\displaystyle \pi _{1,{\textrm {loc}}}}$ is the fundamental group of the intersection of ${\displaystyle B}$ with the set of regular values of f.⁹