Seminar on Perverse Sheaves and Applications, Spring 2017

B.J.J. Moonen, Radboud University

The seminar on Mondays, 15:45 till 17:30 in room HG00.633. Due to holidays, we may occasionally have to skip some weeks. See also the schedule below.

In one of his first published papers, Deligne proved that if f: X → Y is a smooth projective morphism of schemes and ℓ is a prime number that is invertible on Y, the Leray spectral sequence E^{pq} = H^{p}(Y,R^{q}f_{*}**Q**_{ℓ}) ⇒ H^{p+q}(X,**Q**_{ℓ}) degenerates at the E_{2}-level. This means that H^{n}(X,**Q**_{ℓ}) decomposes (non-canonically) as a direct sum of the H^{p}(Y,R^{q}f_{*}**Q**_{ℓ}) with p + q = n. This is an important theorem, and in principle it could form the basis for inductive arguments towards general results about the cohomology of varieties. For such arguments, however, one would need to consider situations where the base variety Y is projective, and then we are confronted with the fact that there are, in some sense, not so many smooth morphisms to a complete base variety. (One is usually confronted with some singular fibres.)

The theory of perverse sheaves, and in particular the so-called decomposition theorem, provides a vast generalization of Deligne’s theorem to a setting where singularities are allowed. To quote MacPherson: *If the only spaces you are interested in are nonsingular manifolds and the only maps you are interested in are fibrations, then perverse sheaves won't matter at all to you. Perverse sheaves involve singularities in an essential way.*

A major inspiration for the abstract theory developed by Beilinson, Bernstein, Deligne and Gabber, was the homology theory (intersection homology) developed by Goresky and MacPherson. On the other hand, for the proofs in the purely algebraic setting, Deligne's theory of weights (as developed, notably, in 'Weil II') plays a crucial role.

The theory of perverse sheaves has found numerous applications and also stands at the basis of later developments, such as Saito’s theory of mixed Hodge modules. But it's not easy to fully digest the theory, and people have continued to look for a better understanding of it, especially in concrete geometric situations. For instance, de Cataldo and Migliorini have obtained a different proof of the decomposition theorem, which is more geometric in flavour.

The main purpose of the seminar is to work through the key notions in the theory, mostly following the seminal paper of Beilinson, Bernstein and Deligne in Astérisque 100, quoted as 'BBD'. At a minimum, we want to reach the decomposition theorem, and gain some understanding of how it is proven. If there is enough energy among the participants, the proposal is to then continue with some talks about the approach by de Cataldo and Migliorini, and to discuss some concrete applications.

There is a tentative programme for the first lectures. Note: I will keep updating and extending this. If you are willing to give one of the talks, let me know. (As soon as the speakers are known, I will include them in the programme.)

We assume some familiarity with the following topics.

- Standard algebraic geometry at the level of Hartshorne’s book.
- Étale cohomology (though some basic notions will be recalled in the first lecture).
- The derived category of an abelian category, and some basic principles of homological algebra. (The notion of a triangulated category will be reviewed in the second lecture.)

Of course, the whole point of running a seminar is that we learn something from it, so we can try to catch up on anything that is relevant to the participants.

Here are links to some very useful texts:

- Faisceaux pervers, by A. Beilinson, A. Bernstein and P. Deligne.
- The stacks project.
- R. MacPherson's notes on Intersection Homology.
- The pro-étale topology for schemes, by B. Bhatt and P. Scholze.
- The Hodge theory of the decomposition theorem (after de Cataldo and Migliorini), by G. Williamson.
- Notes on étale cohomology by Brian Conrad, in case you'd like to see more details than what we can cover in the first lecture.

Date | Speaker | Topic |
---|---|---|

February 13 | Johan Commelin | The étale site of a scheme |

February 20 | Steffen Sagave | Triangulated categories and t-structures |

February 27 | no seminar | |

March 6 | Ben Moonen | The category D_{c}(X,Q_{ℓ}) and the six functors. |

March 13 | Milan Lopuhaä | The perverse t-structure |

March 20 | Salvatore Floccari | The middle (or intermediate) extension functor |

March 27 | Pol van Hoften | Mixed complexes. Notes |

April 3 | Arne Smeets | The weight filtration |

April 10 | Johan Commelin | The decomposition theorem (over finite fields) |

April 17 | no seminar | |

April 24 | no seminar | |

May 1 | no seminar | |

May 8 | ||

May 15 | ||

May 22 | ||

May 29 |