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 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

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

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 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

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

  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. 1 2 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). Société Mathématique de France, Paris. 100.
  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

Pedagogical References

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

Further reading

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