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.^[1]

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 $f:X\to Y$ of relative dimension d between two projective varieties^[2]

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

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

Proofs

The decomposition theorem was first proved by Beilinson, Bernstein, and Deligne.^[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]

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.^[9]

References

^ Conjecture 2.10. of Sergei Gelfand & Robert MacPherson, Verma modules and Schubert cells: A dictionary.
^ 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, S2CID 121086388, Zbl 0159.22501
^ ^a ^b Beilinson, Bernstein & Deligne 1982, Théorème 6.2.10.. NB: To be precise, the reference is for the decomposition.
^ MacPherson 1990, Theorem 1.12. NB: To be precise, the reference is for the decomposition.
^ Beilinson, Bernstein & Deligne 1982, Théorème 6.2.5.
^ Beilinson, Alexander A.; Bernstein, Joseph; Deligne, Pierre (1982). "Faisceaux pervers". Astérisque (in French). 100. Société Mathématique de France, Paris.
^ 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.
^ 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
^ de Cataldo 2015, Theorem 1.4.1.

Survey Articles

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).

Pedagogical References

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