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.

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 → Y {\displaystyle f:X\to Y} of relative dimension d between two projective varieties

− ∪ η i : R d − i f ∗ ( Q ) → ≅ R d + i f ∗ ( Q ) . {\displaystyle -\cup \eta ^{i}:R^{d-i}f_{*}(\mathbb {Q} ){\stackrel {\cong }{\to }}R^{d+i}f_{*}(\mathbb {Q} ).}

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

− ∪ η i : H d − i ( X , Q ) → ≅ H d + i ( X , Q ) . {\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

R f ∗ ( Q ) → ≅ ⨁ i = − d d R d + i f ∗ ( Q ) [ − d − i ] . {\displaystyle 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 ∗ Q {\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.

Decomposition for proper maps

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

p H − i ( R f ∗ Q ) ≅ p H + i ( R f ∗ Q ) , {\displaystyle {}^{p}H^{-i}(Rf_{*}\mathbb {Q} )\cong {}^{p}H^{+i}(Rf_{*}\mathbb {Q} ),}

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

Moreover, there is an isomorphism

R f ∗ I C X ∙ ≅ ⨁ i p H i ( R f ∗ I C X ∙ ) [ − i ] . {\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.

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

Applications of the theorem

Cohomology of a Rational Lefschetz Pencil

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

E 2 l , m = H l ( P 1 ; p H m ( I C X ~ ∙ ( Q ) ) ⇒ I H l + m ( X ~ ; Q ) ≅ H l + m ( X ; Q ) {\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} )}

Local invariant cycle theorem

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

H ∗ ⁡ ( f − 1 ( y ) , Q ) = H ∗ ⁡ ( f − 1 ( B ) , Q ) → H ∗ ⁡ ( f − 1 ( y 0 ) , Q ) π 1 , loc {\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 π 1 , loc {\displaystyle \pi _{1,{\textrm {loc}}}} is the fundamental group of the intersection of B {\displaystyle B} with the set of regular values of f.

Survey Articles

  • de Cataldo, Mark (2015), (PDF), archived from (PDF) on 2015-11-21
  • de Cataldo, Mark; Milgiorini, Luca, (PDF)
  • MacPherson, R. (1990). (PDF).

Pedagogical References

  • Hotta, Ryoshi; Takeuchi, Kiyoshi; Tanisaki, Toshiyuki, D-Modules, Perverse Sheaves, and Representation Theory

Further reading

  • at nLab