Decomposition theorem of Beilinson, Bernstein and Deligne

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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

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Decomposition for smooth proper maps

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The first case of the decomposition theorem arises via the hard Lefschetz theorem which gives isomorphisms, for a smooth proper map   of relative dimension d between two projective varieties[2]

 

Here   is the fundamental class of a hyperplane section,   is the direct image (pushforward) and   is the n-th derived functor of the direct image. This derived functor measures the n-th cohomologies of  , for  . In fact, the particular case when Y is a point, amounts to the isomorphism

 

This hard Lefschetz isomorphism induces canonical isomorphisms

 

Moreover, the sheaves   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

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The decomposition theorem generalizes this fact to the case of a proper, but not necessarily smooth map   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:

 

where   is the total derived functor of   and   is the i-th truncation with respect to the perverse t-structure.

Moreover, there is an isomorphism

 

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   is replaced by the intersection cohomology complex  .[3]

Proofs

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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

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Cohomology of a Rational Lefschetz Pencil

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Consider a rational morphism   from a smooth quasi-projective variety given by  . If we set the vanishing locus of   as   then there is an induced morphism  . We can compute the cohomology of   from the intersection cohomology of   and subtracting off the cohomology from the blowup along  . This can be done using the perverse spectral sequence

 

Local invariant cycle theorem

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Let   be a proper morphism between complex algebraic varieties such that   is smooth. Also, let   be a regular value of   that is in an open ball B centered at  . Then the restriction map

 

is surjective, where   is the fundamental group of the intersection of   with the set of regular values of f.[9]

References

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  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, 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.

Survey Articles

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Pedagogical References

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

Further reading

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