Topology of Algebraic Varieties Learning Seminar

From TorontoMathWiki

A learning seminar on topology of algebraic varieties. We meet on Tuesdays 9:30-11:00 in BA 6180.

Topics We will cover some subset of the following topics: Lefschetz theorems, Hodge decomposition, intersection homology, decomposition theorem, perverse sheaves, mixed Hodge structures, with applications to toric varieties, geometric Satake correspondence, and Ngo's proof of the fundamental Lemma.

References

1. Our main reference is the survey paper "The decomposition theorem, perverse sheaves, and the topology of algebraic maps" by Cataldo and Migliorini, recently published in the Bulletin of the AMS and available here: http://front.math.ucdavis.edu/0712.0349
2. "Intersection homology theory" by Goresky and MacPherson.
3. Introduction to intersection homology theory by Kirwan.
4. Notes on Perverse Sheaves and Vanishing Cycles by David B. Massey http://arxiv.org/abs/math/9908107v2
5. Algebraic Geometry over the complex numbers by Arapura http://www.math.purdue.edu/~dvb/book.html
6. Intersection homology II, by Goresky and MacPherson.

Schedule

1. Jan 12, Smooth projective varieties, cohomology, and Lefschetz theorems (1.1), Joel
2. Jan 19, Families of projective varieties, monodromy, and degeneration of Leray-Serre spectral sequence (1.2) Stephen
3. Jan 26, Intersection homology (topological approach), decomposition and examples (1.3, 1.4) Misha M
4. Feb 2, Intersection homology (topological approach), decomposition and examples #2 (1.3, 1.4) Misha M
5. Feb 9, Cohomology of Sheaves and Derived Categories (1.5) Arthur
6. Feb 23, Cohomology of Sheaves and Derived Categories / IC Sheaves + Perverse Sheaves - Arthur/Omar
7. Mar 2, IC Sheaves + Perverse Sheaves (2.1, 2.2) Omar
8. Mar 9, Perverse Sheaves + Decomposition theorem - Omar/Chris (1.6, 1.8)
9. Mar 16, Decomposition theorem examples - Chris
10. Mar 23, Semismall maps and Springer theory I (4.2) Sergey
11. Mar 30, Semismall maps and Springer theory II (4.2) Sergey
12. April 13, Sheaf to functions and Kazhdan-Lusztig polynomial (4.3,4.4) Brad
13. April 15, Geometric Satake isomorphism (4.5) Bruce
14. April 20, Riemann-Hilbert Correspondence, Dan

Also it would be nice to have a couple of talks on Hodge theory at some point.