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Decomposition theorem of Beilinson, Bernstein and Deligne

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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 f : X → Y {\displaystyle f:X\to Y} of relative dimension d between two projective varieties^[2]

− ∪ 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.

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:^[3][4] 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.^[5]

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} .^[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 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} )}

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.^[9]

1. ^ Conjecture 2.10. of Sergei Gelfand & Robert MacPherson, Verma modules and Schubert cells: A dictionary.
2. ^ Deligne, Pierre (1968), "Théoreme de Lefschetz et critères de dégénérescence de suites spectrales", Publ. Math. Inst. Hautes Études Sci., 35: 107–126, doi:10.1007/BF02698925, hdl:2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/215173, S2CID 121086388, Zbl 0159.22501
3. ^ ^{a b} Beilinson, Bernstein & Deligne 1982, Théorème 6.2.10.. NB: To be precise, the reference is for the decomposition.
4. ^ MacPherson 1990, Theorem 1.12. NB: To be precise, the reference is for the decomposition.
5. ^ Beilinson, Bernstein & Deligne 1982, Théorème 6.2.5.
6. ^ Beilinson, Alexander A.; Bernstein, Joseph; Deligne, Pierre (1982). "Faisceaux pervers". Astérisque (in French). 100. Société Mathématique de France, Paris.
7. ^ de Cataldo, Mark Andrea; Migliorini, Luca (2005). "The Hodge theory of algebraic maps". Annales Scientifiques de l'École Normale Supérieure. 38 (5): 693–750. arXiv:math/0306030. Bibcode:2003math......6030D. doi:10.1016/j.ansens.2005.07.001. S2CID 54046571.
8. ^ de Cataldo, Mark Andrea; Migliorini, Luca (2004), "The Chow motive of semismall resolutions", Math. Res. Lett., 11 (2–3): 151–170, arXiv:math/0204067, doi:10.4310/MRL.2004.v11.n2.a2, MR 2067464, S2CID 53323330
9. ^ de Cataldo 2015, Theorem 1.4.1.

• de Cataldo, Mark (2015), Perverse sheaves and the topology of algebraic varieties Five lectures at the 2015 PCMI (PDF), archived from the original (PDF) on 2015-11-21, retrieved 2017-08-19
• de Cataldo, Mark; Milgiorini, Luca, The Decomposition Theorem, Perverse Sheaves, and the Topology of Algebraic Maps (PDF)
• MacPherson, R. (1990). "Intersection homology and perverse sheaves" (PDF).

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

• BBDG decomposition theorem at nLab

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